SmartRetail: Customer Segmentation for Micro-Targeting

Import Libraries

rm(list=ls())
library(plotly)

## Loading required package: ggplot2

## 
## Attaching package: 'plotly'

## The following object is masked from 'package:ggplot2':
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## The following object is masked from 'package:stats':
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library(ROCR)
library(readr); library(dplyr); library(tidyr); library(ggplot2)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
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## The following objects are masked from 'package:base':
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library(rmarkdown)
library(GGally)

## Registered S3 method overwritten by 'GGally':
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library(tree)
library(DataExplorer)
library(factoextra)

## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa

library(randomForest)

## randomForest 4.7-1.2

## Type rfNews() to see new features/changes/bug fixes.

## 
## Attaching package: 'randomForest'

## The following object is masked from 'package:dplyr':
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library(clValid)

## Loading required package: cluster

Data Description

Load Data

The data set is unstructured and we want to import data into a tidy format (i.e. dataframe)

# To input this unstructured file into R we use read_delim 
Customers <- read_delim("/Users/ogheneatoma/Documents/ST309 final project - SmartRetail/data/marketing_campaign.csv", delim = "\t")

## Rows: 2240 Columns: 29
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: "\t"
## chr  (3): Education, Marital_Status, Dt_Customer
## dbl (26): ID, Year_Birth, Income, Kidhome, Teenhome, Recency, MntWines, MntF...
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

# To show # of observations and features in this data set
dim(Customers)

## [1] 2240   29

Now we have a tidy data set with each feature forming a column and each value having its own cell

Basic examination of the dataset

Column names:

names(Customers)

##  [1] "ID"                  "Year_Birth"          "Education"          
##  [4] "Marital_Status"      "Income"              "Kidhome"            
##  [7] "Teenhome"            "Dt_Customer"         "Recency"            
## [10] "MntWines"            "MntFruits"           "MntMeatProducts"    
## [13] "MntFishProducts"     "MntSweetProducts"    "MntGoldProds"       
## [16] "NumDealsPurchases"   "NumWebPurchases"     "NumCatalogPurchases"
## [19] "NumStorePurchases"   "NumWebVisitsMonth"   "AcceptedCmp3"       
## [22] "AcceptedCmp4"        "AcceptedCmp5"        "AcceptedCmp1"       
## [25] "AcceptedCmp2"        "Complain"            "Z_CostContact"      
## [28] "Z_Revenue"           "Response"

Unique values in each feature

nunique <- function(x) length(unique(x))
nunique_counts <- sapply(Customers, nunique)
nunique_counts

##                  ID          Year_Birth           Education      Marital_Status 
##                2240                  59                   5                   8 
##              Income             Kidhome            Teenhome         Dt_Customer 
##                1975                   3                   3                 663 
##             Recency            MntWines           MntFruits     MntMeatProducts 
##                 100                 776                 158                 558 
##     MntFishProducts    MntSweetProducts        MntGoldProds   NumDealsPurchases 
##                 182                 177                 213                  15 
##     NumWebPurchases NumCatalogPurchases   NumStorePurchases   NumWebVisitsMonth 
##                  15                  14                  14                  16 
##        AcceptedCmp3        AcceptedCmp4        AcceptedCmp5        AcceptedCmp1 
##                   2                   2                   2                   2 
##        AcceptedCmp2            Complain       Z_CostContact           Z_Revenue 
##                   2                   2                   1                   1 
##            Response 
##                   2

Summary statistics

summary(Customers)

##        ID          Year_Birth    Education         Marital_Status    
##  Min.   :    0   Min.   :1893   Length:2240        Length:2240       
##  1st Qu.: 2828   1st Qu.:1959   Class :character   Class :character  
##  Median : 5458   Median :1970   Mode  :character   Mode  :character  
##  Mean   : 5592   Mean   :1969                                        
##  3rd Qu.: 8428   3rd Qu.:1977                                        
##  Max.   :11191   Max.   :1996                                        
##                                                                      
##      Income          Kidhome          Teenhome      Dt_Customer       
##  Min.   :  1730   Min.   :0.0000   Min.   :0.0000   Length:2240       
##  1st Qu.: 35303   1st Qu.:0.0000   1st Qu.:0.0000   Class :character  
##  Median : 51382   Median :0.0000   Median :0.0000   Mode  :character  
##  Mean   : 52247   Mean   :0.4442   Mean   :0.5062                     
##  3rd Qu.: 68522   3rd Qu.:1.0000   3rd Qu.:1.0000                     
##  Max.   :666666   Max.   :2.0000   Max.   :2.0000                     
##  NA's   :24                                                           
##     Recency         MntWines         MntFruits     MntMeatProducts
##  Min.   : 0.00   Min.   :   0.00   Min.   :  0.0   Min.   :   0   
##  1st Qu.:24.00   1st Qu.:  23.75   1st Qu.:  1.0   1st Qu.:  16   
##  Median :49.00   Median : 173.50   Median :  8.0   Median :  67   
##  Mean   :49.11   Mean   : 303.94   Mean   : 26.3   Mean   : 167   
##  3rd Qu.:74.00   3rd Qu.: 504.25   3rd Qu.: 33.0   3rd Qu.: 232   
##  Max.   :99.00   Max.   :1493.00   Max.   :199.0   Max.   :1725   
##                                                                   
##  MntFishProducts  MntSweetProducts  MntGoldProds    NumDealsPurchases
##  Min.   :  0.00   Min.   :  0.00   Min.   :  0.00   Min.   : 0.000   
##  1st Qu.:  3.00   1st Qu.:  1.00   1st Qu.:  9.00   1st Qu.: 1.000   
##  Median : 12.00   Median :  8.00   Median : 24.00   Median : 2.000   
##  Mean   : 37.53   Mean   : 27.06   Mean   : 44.02   Mean   : 2.325   
##  3rd Qu.: 50.00   3rd Qu.: 33.00   3rd Qu.: 56.00   3rd Qu.: 3.000   
##  Max.   :259.00   Max.   :263.00   Max.   :362.00   Max.   :15.000   
##                                                                      
##  NumWebPurchases  NumCatalogPurchases NumStorePurchases NumWebVisitsMonth
##  Min.   : 0.000   Min.   : 0.000      Min.   : 0.00     Min.   : 0.000   
##  1st Qu.: 2.000   1st Qu.: 0.000      1st Qu.: 3.00     1st Qu.: 3.000   
##  Median : 4.000   Median : 2.000      Median : 5.00     Median : 6.000   
##  Mean   : 4.085   Mean   : 2.662      Mean   : 5.79     Mean   : 5.317   
##  3rd Qu.: 6.000   3rd Qu.: 4.000      3rd Qu.: 8.00     3rd Qu.: 7.000   
##  Max.   :27.000   Max.   :28.000      Max.   :13.00     Max.   :20.000   
##                                                                          
##   AcceptedCmp3      AcceptedCmp4      AcceptedCmp5      AcceptedCmp1    
##  Min.   :0.00000   Min.   :0.00000   Min.   :0.00000   Min.   :0.00000  
##  1st Qu.:0.00000   1st Qu.:0.00000   1st Qu.:0.00000   1st Qu.:0.00000  
##  Median :0.00000   Median :0.00000   Median :0.00000   Median :0.00000  
##  Mean   :0.07277   Mean   :0.07455   Mean   :0.07277   Mean   :0.06429  
##  3rd Qu.:0.00000   3rd Qu.:0.00000   3rd Qu.:0.00000   3rd Qu.:0.00000  
##  Max.   :1.00000   Max.   :1.00000   Max.   :1.00000   Max.   :1.00000  
##                                                                         
##   AcceptedCmp2        Complain        Z_CostContact   Z_Revenue 
##  Min.   :0.00000   Min.   :0.000000   Min.   :3     Min.   :11  
##  1st Qu.:0.00000   1st Qu.:0.000000   1st Qu.:3     1st Qu.:11  
##  Median :0.00000   Median :0.000000   Median :3     Median :11  
##  Mean   :0.01339   Mean   :0.009375   Mean   :3     Mean   :11  
##  3rd Qu.:0.00000   3rd Qu.:0.000000   3rd Qu.:3     3rd Qu.:11  
##  Max.   :1.00000   Max.   :1.000000   Max.   :3     Max.   :11  
##                                                                 
##     Response     
##  Min.   :0.0000  
##  1st Qu.:0.0000  
##  Median :0.0000  
##  Mean   :0.1491  
##  3rd Qu.:0.0000  
##  Max.   :1.0000  
## 

Data Cleaning

Missing Values

plot_missing(Customers)

Customers <- na.omit(Customers)

There are 24 observations dropped in the data set because of missing values.

Duplicate Values

Check if there are duplicated data on the same customer characterised by their ID.

duplicates <- Customers$ID[duplicated(Customers$ID)]
duplicates

## numeric(0)

No duplicates found.

Outliers

Based on the outliers found in the summary statistics, we will examine

par(mfrow = c(1, 2))
hist(2021-Customers$Year_Birth)
hist(Customers$Income)

Based on the histogram plots, we can see the majority of individuals are aged below 80 and have an income level below 100,000. Hence we drop the unusual values.

Removing the outliers

Customers_cleaned <- subset(Customers, 2021-Customers$Year_Birth < 80 & Income < 100000)

This drops another 24 observations.

Feature Engineering/Data Transformation

# Age category for each customer assuming the data was collected in 2021
Customers_cleaned$Age <- 2021 - Customers_cleaned$Year_Birth

# Cut into differnet age groups by generations for our analysis  
Customers_cleaned$AgeCategory <- cut(Customers_cleaned$Age, c(0, 40, 56, 78), c('<40', '41-56', '>57')) 

# Number of Children
Customers_cleaned$NumChildren <- Customers_cleaned$Kidhome + Customers_cleaned$Teenhome

# Total spending
Customers_cleaned$Spending <- Customers_cleaned$MntWines + Customers_cleaned$MntFruits + Customers_cleaned$MntMeatProducts + Customers_cleaned$MntFishProducts + Customers_cleaned$MntSweetProducts + Customers_cleaned$MntGoldProds

# Log transformation of highly-skewed spending variables
Customers_cleaned <- Customers_cleaned %>%
  mutate(
    log_Wines = log(1+MntWines),
    log_Fruits = log(1+MntFruits),
    log_MeatProducts = log(1+MntMeatProducts),
    log_FishProducts = log(1+MntFishProducts),
    log_SweetProducts = log(1+MntSweetProducts),
    log_GoldProds = log(1+MntGoldProds),
    log_Spending = log(1+Spending) 
  )

# Relationship
Customers_cleaned$Relationship <- ifelse(Customers_cleaned$Marital_Status %in% c("Married", "Together"), 1, 0)
Customers_cleaned$Relationship <- factor(Customers_cleaned$Relationship, levels = c(0, 1), labels = c("Not Partnered", "Partnered"))

# Education
Education <- c(Basic = "Bachelors", '2n Cycle' = "Bachelors", Graduation = "Graduate", Master = "Masters", PhD = "PhD")
Customers_cleaned$Education <- as.character(Education[Customers_cleaned$Education])
Customers_cleaned$Education <- factor(Customers_cleaned$Education)

# Number of years customers joined 
Dt_Customer <- as.Date(Customers_cleaned$Dt_Customer, format = "%d-%m-%Y")
Year_Customer <- as.numeric(format(Dt_Customer, "%Y"))
Customers_cleaned$YearsJoined <- 2021 - Year_Customer

# Number of accepted campaigns out of 6 in total 
Customers_cleaned$TotalAcceptedCmp <- Customers_cleaned$AcceptedCmp1 + Customers_cleaned$AcceptedCmp2 + Customers_cleaned$AcceptedCmp3 + Customers_cleaned$AcceptedCmp4 + Customers_cleaned$AcceptedCmp5 + Customers_cleaned$Response

# Remove redundant columns
Customers_cleaned <- subset(Customers_cleaned, select = -c(ID, Z_CostContact, Z_Revenue, Year_Birth, Marital_Status, Dt_Customer, Teenhome, Kidhome))

Convert all variables into numerical using label encoding.

# Examine data types of the columns 
str(Customers_cleaned)

## tibble [2,198 × 35] (S3: tbl_df/tbl/data.frame)
##  $ Education          : Factor w/ 4 levels "Bachelors","Graduate",..: 2 2 2 2 4 3 2 4 4 4 ...
##  $ Income             : num [1:2198] 58138 46344 71613 26646 58293 ...
##  $ Recency            : num [1:2198] 58 38 26 26 94 16 34 32 19 68 ...
##  $ MntWines           : num [1:2198] 635 11 426 11 173 520 235 76 14 28 ...
##  $ MntFruits          : num [1:2198] 88 1 49 4 43 42 65 10 0 0 ...
##  $ MntMeatProducts    : num [1:2198] 546 6 127 20 118 98 164 56 24 6 ...
##  $ MntFishProducts    : num [1:2198] 172 2 111 10 46 0 50 3 3 1 ...
##  $ MntSweetProducts   : num [1:2198] 88 1 21 3 27 42 49 1 3 1 ...
##  $ MntGoldProds       : num [1:2198] 88 6 42 5 15 14 27 23 2 13 ...
##  $ NumDealsPurchases  : num [1:2198] 3 2 1 2 5 2 4 2 1 1 ...
##  $ NumWebPurchases    : num [1:2198] 8 1 8 2 5 6 7 4 3 1 ...
##  $ NumCatalogPurchases: num [1:2198] 10 1 2 0 3 4 3 0 0 0 ...
##  $ NumStorePurchases  : num [1:2198] 4 2 10 4 6 10 7 4 2 0 ...
##  $ NumWebVisitsMonth  : num [1:2198] 7 5 4 6 5 6 6 8 9 20 ...
##  $ AcceptedCmp3       : num [1:2198] 0 0 0 0 0 0 0 0 0 1 ...
##  $ AcceptedCmp4       : num [1:2198] 0 0 0 0 0 0 0 0 0 0 ...
##  $ AcceptedCmp5       : num [1:2198] 0 0 0 0 0 0 0 0 0 0 ...
##  $ AcceptedCmp1       : num [1:2198] 0 0 0 0 0 0 0 0 0 0 ...
##  $ AcceptedCmp2       : num [1:2198] 0 0 0 0 0 0 0 0 0 0 ...
##  $ Complain           : num [1:2198] 0 0 0 0 0 0 0 0 0 0 ...
##  $ Response           : num [1:2198] 1 0 0 0 0 0 0 0 1 0 ...
##  $ Age                : num [1:2198] 64 67 56 37 40 54 50 36 47 71 ...
##  $ AgeCategory        : Factor w/ 3 levels "<40","41-56",..: 3 3 2 1 1 2 2 1 2 3 ...
##  $ NumChildren        : num [1:2198] 0 2 0 1 1 1 1 1 1 2 ...
##  $ Spending           : num [1:2198] 1617 27 776 53 422 ...
##  $ log_Wines          : num [1:2198] 6.46 2.48 6.06 2.48 5.16 ...
##  $ log_Fruits         : num [1:2198] 4.489 0.693 3.912 1.609 3.784 ...
##  $ log_MeatProducts   : num [1:2198] 6.3 1.95 4.85 3.04 4.78 ...
##  $ log_FishProducts   : num [1:2198] 5.15 1.1 4.72 2.4 3.85 ...
##  $ log_SweetProducts  : num [1:2198] 4.489 0.693 3.091 1.386 3.332 ...
##  $ log_GoldProds      : num [1:2198] 4.49 1.95 3.76 1.79 2.77 ...
##  $ log_Spending       : num [1:2198] 7.39 3.33 6.66 3.99 6.05 ...
##  $ Relationship       : Factor w/ 2 levels "Not Partnered",..: 1 1 2 2 2 2 1 2 2 2 ...
##  $ YearsJoined        : num [1:2198] 9 7 8 7 7 8 9 8 8 7 ...
##  $ TotalAcceptedCmp   : num [1:2198] 1 0 0 0 0 0 0 0 1 1 ...
##  - attr(*, "na.action")= 'omit' Named int [1:24] 11 28 44 49 59 72 91 92 93 129 ...
##   ..- attr(*, "names")= chr [1:24] "11" "28" "44" "49" ...

# Identify categorical columns
categorical_cols <- sapply(Customers_cleaned, is.factor)

# Apply label encoding to categorical columns
Customers_cleaned[categorical_cols] <- lapply(Customers_cleaned[categorical_cols], as.numeric)

Subset the data frame used for PCA

Customers_PCA <- subset(Customers_cleaned, select = -c(Spending,MntWines,MntFruits,MntMeatProducts,MntFishProducts,MntSweetProducts,MntGoldProds,AcceptedCmp1,AcceptedCmp2,AcceptedCmp3,AcceptedCmp4,AcceptedCmp5,Complain,Response, AgeCategory))

Data Analysis

EDA

Pick out on some key features for correlation analysis.

plot_correlation(Customers_cleaned)

plot_histogram(Customers_cleaned)

PCA

Now the columns of the data set contain the following variables.

names(Customers_PCA)

##  [1] "Education"           "Income"              "Recency"            
##  [4] "NumDealsPurchases"   "NumWebPurchases"     "NumCatalogPurchases"
##  [7] "NumStorePurchases"   "NumWebVisitsMonth"   "Age"                
## [10] "NumChildren"         "log_Wines"           "log_Fruits"         
## [13] "log_MeatProducts"    "log_FishProducts"    "log_SweetProducts"  
## [16] "log_GoldProds"       "log_Spending"        "Relationship"       
## [19] "YearsJoined"         "TotalAcceptedCmp"

We examine the mean and variances of different variables

apply(Customers_PCA, 2, mean)

##           Education              Income             Recency   NumDealsPurchases 
##        2.480437e+00        5.148159e+04        4.900773e+01        2.324841e+00 
##     NumWebPurchases NumCatalogPurchases   NumStorePurchases   NumWebVisitsMonth 
##        4.084167e+00        2.633303e+00        5.815742e+00        5.345314e+00 
##                 Age         NumChildren           log_Wines          log_Fruits 
##        5.207188e+01        9.517743e-01        4.683589e+00        2.244250e+00 
##    log_MeatProducts    log_FishProducts   log_SweetProducts       log_GoldProds 
##        4.130463e+00        2.540630e+00        2.244045e+00        3.126917e+00 
##        log_Spending        Relationship         YearsJoined    TotalAcceptedCmp 
##        5.623144e+00        1.645587e+00        7.972247e+00        4.440400e-01

apply(Customers_PCA, 2, var)

##           Education              Income             Recency   NumDealsPurchases 
##        9.088115e-01        4.230185e+08        8.364920e+02        3.554877e+00 
##     NumWebPurchases NumCatalogPurchases   NumStorePurchases   NumWebVisitsMonth 
##        7.255999e+00        7.765336e+00        1.044714e+01        5.816526e+00 
##                 Age         NumChildren           log_Wines          log_Fruits 
##        1.363344e+02        5.602586e-01        3.244145e+00        2.458667e+00 
##    log_MeatProducts    log_FishProducts   log_SweetProducts       log_GoldProds 
##        2.413958e+00        2.747391e+00        2.524857e+00        1.644474e+00 
##        log_Spending        Relationship         YearsJoined    TotalAcceptedCmp 
##        2.149672e+00        2.289086e-01        4.712367e-01        7.777046e-01

PCA with standardised variables

PCA <- prcomp(x = Customers_PCA, scale = TRUE)
Table_PCA <- rbind(PCA$rotation, summary(PCA)$importance)
knitr::kable(Table_PCA, digits = 4, align = 'c')

	PC1	PC2	PC3	PC4	PC5	PC6	PC7	PC8	PC9	PC10	PC11	PC12	PC13	PC14	PC15	PC16	PC17	PC18	PC19	PC20
Education	0.0099	0.2065	-0.5445	-0.1559	0.1085	-0.2216	-0.1056	0.4948	0.0411	0.4625	0.1228	-0.2081	0.1508	0.0390	0.0357	-0.0467	0.1170	0.0324	-0.0546	-0.0120
Income	0.3058	-0.0130	-0.2373	0.0273	-0.0125	0.0062	0.0785	-0.0277	-0.1706	-0.2592	-0.0195	-0.1899	-0.1028	-0.0239	0.0144	0.0783	-0.3221	0.5816	-0.4992	-0.0278
Recency	0.0068	0.0058	0.0537	0.4368	0.6357	-0.5579	0.2359	-0.1628	0.0379	0.0114	-0.0464	0.0174	0.0558	-0.0055	0.0316	0.0091	0.0166	0.0137	-0.0016	0.0010
NumDealsPurchases	-0.0104	0.5535	0.1884	0.1225	-0.0611	0.1792	0.2604	0.0198	-0.2750	0.3564	-0.0283	0.4273	0.1076	-0.0857	-0.0196	0.3061	0.0176	0.1707	-0.0974	0.0317
NumWebPurchases	0.2254	0.3456	0.0570	-0.0440	-0.0460	-0.0024	0.1143	0.0210	0.5452	-0.1734	-0.2162	-0.1865	-0.3156	0.0654	-0.0310	0.2502	0.4719	0.0143	-0.0894	0.0416
NumCatalogPurchases	0.2883	-0.0400	-0.0817	-0.0631	0.0310	-0.0747	0.0025	-0.0931	-0.2272	0.1560	0.1442	0.3138	-0.5472	0.0117	-0.2987	-0.4736	0.2741	0.0679	0.0183	0.0312
NumStorePurchases	0.2760	0.0952	-0.0613	0.1198	-0.0464	0.0541	0.0156	0.2280	-0.0446	-0.4049	-0.2024	0.2737	0.5321	-0.0032	0.1097	-0.4168	0.2877	-0.0280	-0.0678	0.0202
NumWebVisitsMonth	-0.2080	0.3490	0.2718	-0.2125	0.0135	-0.1036	-0.0231	-0.0196	0.3465	0.1250	-0.2421	-0.0155	-0.0225	-0.0219	-0.0248	-0.5513	-0.4183	0.1254	-0.1139	-0.0180
Age	0.0560	0.2103	-0.3554	0.2673	0.0829	0.2143	-0.5862	-0.5450	0.1302	0.1267	-0.1079	0.0727	0.0925	0.0206	-0.0019	-0.0034	-0.0075	-0.0333	-0.0373	0.0008
NumChildren	-0.1881	0.3819	-0.0579	0.2206	-0.0329	0.1862	0.2114	-0.1085	-0.4217	-0.1310	0.0262	-0.5757	-0.1119	0.0523	0.0593	-0.2670	0.1006	-0.1977	0.0750	-0.0507
log_Wines	0.3035	0.2296	-0.1814	-0.0297	0.0016	-0.0413	0.0576	0.0507	0.0247	-0.1875	0.0289	0.0269	-0.0594	-0.0208	0.0129	0.0618	-0.3677	0.0669	0.6619	0.4367
log_Fruits	0.2709	-0.1363	0.2068	0.0901	0.0083	0.0890	-0.0546	0.0178	-0.0075	0.2371	-0.1709	-0.3440	0.1942	-0.6599	-0.3819	-0.0271	0.0736	0.0763	0.1005	0.0318
log_MeatProducts	0.3327	0.0417	0.0017	0.0007	0.0068	-0.0278	0.0307	0.0760	-0.0446	0.0424	-0.0396	0.0287	-0.0987	-0.0675	-0.0013	0.0606	-0.3094	-0.7196	-0.4329	0.2344
log_FishProducts	0.2695	-0.1488	0.2141	0.1147	-0.0118	0.0889	-0.0895	0.0444	-0.0358	0.3071	-0.1171	-0.0792	-0.1842	-0.0560	0.8020	-0.1110	0.0568	0.1016	0.0718	0.0370
log_SweetProducts	0.2682	-0.1394	0.2127	0.1034	-0.0303	0.0589	-0.0170	0.0436	-0.0954	0.2646	-0.2998	-0.1845	0.1568	0.7232	-0.2875	0.0194	-0.0603	0.0527	0.0725	0.0283
log_GoldProds	0.2452	0.0960	0.2511	-0.0175	0.0071	0.0739	0.0515	-0.1863	0.2328	0.0600	0.7900	-0.1388	0.2740	0.1196	0.0185	-0.1113	0.0374	0.0506	-0.0916	0.0936
log_Spending	0.3395	0.1317	-0.0207	-0.0244	-0.0030	-0.0243	0.0374	0.0231	0.0425	-0.0388	0.0567	0.0498	-0.0278	-0.0213	0.0079	0.0554	-0.2085	-0.1506	0.2244	-0.8567
Relationship	-0.0087	0.0305	-0.0121	0.3209	-0.7386	-0.5750	-0.0405	-0.1074	0.0073	0.0603	0.0311	-0.0156	0.0279	-0.0284	0.0040	-0.0051	-0.0005	0.0026	-0.0029	0.0038
YearsJoined	0.0454	0.2580	0.3823	-0.2454	0.1398	-0.3168	-0.6075	0.1194	-0.3588	-0.2213	0.0528	-0.0787	-0.0273	0.0113	0.0038	0.1559	0.0934	0.0287	-0.0160	0.0105
TotalAcceptedCmp	0.1269	-0.0099	-0.1227	-0.6218	-0.0252	-0.2209	0.2584	-0.5326	-0.1693	0.0969	-0.1893	-0.0648	0.2478	-0.0095	0.1363	0.0368	0.1395	-0.0509	0.0187	0.0194
Standard deviation	2.8356	1.4195	1.2694	1.0745	1.0039	0.9764	0.9098	0.8908	0.7654	0.7058	0.6740	0.6399	0.5985	0.5555	0.5454	0.4990	0.4822	0.3183	0.2892	0.1316
Proportion of Variance	0.4020	0.1007	0.0806	0.0577	0.0504	0.0477	0.0414	0.0397	0.0293	0.0249	0.0227	0.0205	0.0179	0.0154	0.0149	0.0124	0.0116	0.0051	0.0042	0.0009
Cumulative Proportion	0.4020	0.5028	0.5834	0.6411	0.6915	0.7391	0.7805	0.8202	0.8495	0.8744	0.8971	0.9176	0.9355	0.9509	0.9658	0.9783	0.9899	0.9950	0.9991	1.0000

par(mfrow=c(1,1))
plot(Table_PCA['Proportion of Variance',], type = 'o', lwd = 5, col = 'blue', main = 'PC proportions of total variance', xlab = 'PC', ylab = 'Proportion of variance', axes = FALSE)
axis(1, 1:22)
axis(2)

Subset data based on chosen principle components

nf <- 3
Customers_clust <- as.data.frame(PCA$x[, 1:nf])
Customers_clust

##                PC1           PC2          PC3
## 1     3.9652558507  0.6607660795  1.591087289
## 2    -3.4722319455 -0.5637275979 -1.280295584
## 3     2.7525441587 -0.5272958013  0.346571407
## 4    -2.7696208652 -1.3743700228  0.358376316
## 5     0.9804712402  0.3991332033 -0.358985060
## 6     1.3043261953  0.7752718737 -0.695340025
## 7     1.7635014298  0.9867400903  1.789659077
## 8    -1.6775500664  0.5132352056 -0.046602800
## 9    -3.3100178001  0.0092184102 -0.927981220
## 10   -5.2371345187  2.1713880533 -0.630846939
## 11   -2.7583928842 -1.2670179871  2.639190479
## 12    3.0727704667 -1.7675497202  0.389638943
## 13   -0.6628823181  1.7770652767 -1.160078604
## 14   -3.1233231557 -1.4913145552  2.000037785
## 15    4.3490029545  0.4297254486 -1.743673033
## 16   -2.4418583812  0.8571300131  1.337978396
## 17    0.1459414193  0.1781602009  0.948443732
## 18    3.2807940346  1.9827950135 -1.033565728
## 19   -1.3813658331 -1.2117437717  3.112212179
## 20    0.0861094993 -0.5663084787  2.151368758
## 21    0.3948656122  1.8400223737  0.507799695
## 22   -0.2380085608  2.5147646708 -1.974358636
## 23    0.7506349783  0.7904782225 -2.606283209
## 24   -0.0400413525  2.6691918008  1.031036764
## 25   -2.0793564504 -1.2896012972  1.295254019
## 26   -1.0689270439  1.4161170214  0.626544647
## 27   -1.2204921389 -1.4192669914  0.802735769
## 28    3.8320679761 -0.8615875584 -2.189351919
## 29   -3.2703833477 -1.8318789285 -0.040975723
## 30    0.7149510517 -1.4444126008 -0.033144327
## 31   -2.5362109581  0.1216694939 -0.268080246
## 32    0.2639791612  2.5550996165 -0.061173080
## 33    2.7759716973 -0.2836910516 -0.671063804
## 34   -3.0298300683  0.5323407943 -1.568173770
## 35    1.9512811569  1.4928263035 -1.646633103
## 36   -2.3616482565 -0.5448546901  2.127933991
## 37   -3.1067010992 -1.0000362343 -0.903058769
## 38    3.0366948807  0.5024741609 -1.315797012
## 39    3.5176639598 -0.2477920599  1.128202946
## 40   -3.4311141848 -0.1808012065  0.408315071
## 41   -3.8438867784 -0.7304714598 -0.058717800
## 42   -3.6090983316 -0.4405422702  0.263005781
## 43    3.3168335607 -2.6323457754  0.326287835
## 44   -1.8283387565 -0.9710529817  3.989211939
## 45   -2.7970650449 -0.3751827533  0.883062861
## 46    2.3199647051  4.2715463838  0.319473409
## 47    1.6994362337 -1.2208635346 -0.993822941
## 48    4.4979383887 -0.2173494688 -0.293133029
## 49   -2.5654134179 -0.8756450957  0.799293114
## 50    4.7962525906 -2.2585565999 -0.327759179
## 51    1.3023736277  1.7083868510  0.848313769
## 52    5.5294531310 -0.7422988021 -0.564586793
## 53    3.5742952709 -1.3039524227  0.241890829
## 54   -3.4675245542 -2.1046530642 -0.440502698
## 55   -0.7574797751  1.8736213376 -2.010978198
## 56    3.9284316382 -1.4812804902 -0.890546150
## 57    1.1613315592  1.0828610617  0.411106976
## 58    1.9430305829  1.9063147534  0.463937456
## 59    0.1029854272  1.2582772758 -2.818562120
## 60    2.9267132792 -0.6690004453  0.032050570
## 61   -3.2898821931 -1.2778443413  0.774527052
## 62   -5.3049942725 -0.7696054380 -1.353404816
## 63    4.3204903088 -0.3937320821 -0.083791812
## 64    2.1321285874 -0.1667907722  0.059107453
## 65    1.9493603831  2.1129303688  0.951018857
## 66    3.2150680577 -0.1169948105 -0.182887381
## 67    3.1168597406  1.6718619363  1.089078859
## 68    0.2722197198  3.0737577545 -0.945189855
## 69   -1.5415584479 -2.2350892518 -0.816528730
## 70   -4.1005463925 -1.1939571037  0.683086167
## 71    3.4672052838 -0.1674536762  0.294530419
## 72    4.6710489370 -1.1824525131  0.914396606
## 73   -3.8032017196 -2.5601412470  0.608395229
## 74    0.2985519516  1.6002732453 -0.654841405
## 75   -1.5335110931  0.7197432338  2.065867208
## 76   -4.2985570264 -1.0132464757  0.046364114
## 77   -2.5645026047 -0.0948997355  1.538538623
## 78    0.7149510517 -1.4444126008 -0.033144327
## 79    3.7073531223 -2.4271159647 -0.295169061
## 80   -2.8814013558 -0.0809374860  0.280798966
## 81   -2.8073961246  1.9759029336 -0.925877777
## 82   -0.2125333784  1.1994083583 -1.110535999
## 83    3.9888325987 -1.8923549644  0.054106400
## 84   -0.9666111741 -1.3506620426 -1.408350732
## 85   -3.5230685385 -1.0314564334 -1.172501864
## 86   -3.5619491751 -0.1141971477 -2.065962887
## 87   -2.9115198410 -1.7914938980  0.432485923
## 88   -0.2184611183  2.6220984515  0.097134421
## 89   -3.3217907419  0.2104560599 -0.397116953
## 90    3.4943581404 -0.7449525808 -0.318276869
## 91    1.1958287650 -0.7837369260  0.153094497
## 92   -2.8453463206 -0.1177074887  0.313069401
## 93   -3.1937742254 -0.1343205436 -2.162555348
## 94    3.9785206616  0.0759014664  1.383159346
## 95    2.9459476733 -0.0417911254 -1.054892748
## 96    3.7278268377 -2.1677911018 -1.279785478
## 97   -3.6299149037 -1.8375179393  1.341828645
## 98   -1.9800754070  1.5501458393 -0.317809383
## 99    2.5144881176  1.2396861196  1.348237705
## 100  -1.6006871168  0.1649056599  1.283658605
## 101   2.7632430131 -0.7336668036 -0.249508904
## 102   3.2665328169 -0.5877542262  0.473173948
## 103   2.6700828420  0.6013339036  0.738083362
## 104  -0.3670268129  1.7086521052 -1.889731798
## 105   3.4761069715 -1.4352659426  0.060937159
## 106  -0.4904133352  2.7655692723 -0.259865613
## 107  -2.3547512902  0.5408185017 -0.495907591
## 108   4.1222850381 -2.5421166383 -0.876024021
## 109   1.0979349262  1.0240563978  3.564964851
## 110  -2.5219242660 -0.4107839665  0.954382074
## 111  -3.3371827471 -1.1514004129  0.769039916
## 112   2.1451019118  1.0065526156 -0.077088803
## 113  -2.2561771054 -0.3849636826 -1.263255473
## 114  -4.0176397646 -0.1571722016  0.327151581
## 115  -3.3322778244 -1.0753349790  0.335755918
## 116   2.3797426127 -1.5797385704  0.917150833
## 117   3.5362847327 -1.7887947460  0.327957442
## 118  -3.0057138303  0.6293489413 -2.436838952
## 119   1.4382019094  1.5337263522 -2.215585317
## 120   1.6441636057  0.6889222964  0.766183334
## 121   0.0253696598  3.5204700813 -0.488007154
## 122   2.6944603803 -1.1090516375  0.125104110
## 123  -1.7958720798  2.8591966940 -1.071377860
## 124   2.7006178135 -0.7157587048 -2.131963312
## 125  -4.0618226731 -0.9608663264  2.565677154
## 126  -2.9763093280  2.0661494793 -1.285996507
## 127  -2.9324795256 -0.3322271912 -1.796261601
## 128  -3.4165423452  0.5689530638 -1.172016700
## 129   4.0784568572 -0.9895688269  0.422384119
## 130   1.1746004209  0.6015153156 -0.586947287
## 131   3.9737959275 -0.9543866291 -0.179411112
## 132   2.2818062962  1.1803550102 -0.429980446
## 133  -0.1179119742 -0.3855062806  2.204727670
## 134   0.9382327607  1.2875400421  1.610124738
## 135  -2.4321280385 -0.9431313519  1.497916514
## 136  -4.0552091495 -0.9594306302  0.378099088
## 137  -1.7414615045  2.9738334655  2.346251288
## 138  -0.6162592639 -2.0791388398 -0.999270200
## 139   0.8930438021 -0.1977835646 -1.775555645
## 140   0.7757600129  0.6920155949 -0.063764297
## 141   0.4287781900  1.2977573418  2.417264174
## 142  -1.9093338872 -0.9568516760  1.237849585
## 143   0.9684153127  2.6390515242 -1.930000954
## 144   3.3209569738  0.2132125984 -0.886809551
## 145  -0.5909894468  0.8162827002 -0.203732153
## 146  -4.1726674745 -0.7920528403 -0.872310755
## 147  -0.3579880768 -1.7435022247  0.105974556
## 148   3.4867547622 -1.4181245283 -0.393747588
## 149  -5.0226742219  0.5218709224 -1.260618478
## 150   3.3259832577  1.8124611784 -1.044649135
## 151  -2.3297664739 -0.0375841735  0.762149843
## 152   2.4464508082 -0.6316359332 -1.692925980
## 153  -2.5020862933  0.9943284167  0.300740300
## 154   2.1215367148 -1.2024143750 -0.284846667
## 155  -0.5753197159 -0.9634341009 -1.160433377
## 156   3.2767512218 -2.3620560576  1.345606563
## 157  -2.0339434607 -0.2681048543  0.038629901
## 158  -3.5037710468 -0.5160380834 -0.252288397
## 159  -2.7638575173 -0.7072509780  1.484792575
## 160  -1.7290807210 -0.3025237911 -0.566312362
## 161  -3.0336766973  0.5656765295 -2.952222761
## 162  -2.0687860172 -0.9215876780 -1.287952848
## 163   4.0260788633  1.1059503573 -0.075157626
## 164   4.6859105458  0.5022075197 -1.029454895
## 165  -3.8973142900  0.4599550682 -2.111521209
## 166  -4.6686882460 -1.1126314299 -1.755788396
## 167   3.4943581404 -0.7449525808 -0.318276869
## 168  -1.3475272866 -0.4349155637 -1.774383538
## 169  -2.7543853965 -1.4475533379  1.588275231
## 170   3.3696008563 -0.1029516327 -1.206057267
## 171  -3.3507601479 -0.9868991025 -1.015662939
## 172  -2.7704369342 -2.2869429481  0.708948897
## 173  -3.1677564692 -0.5983518326  1.693300507
## 174  -3.9036341629 -0.5361973340 -2.267217736
## 175   0.8762051640  0.8285550249  0.067409608
## 176   3.8652564840 -1.6593986579 -0.023690609
## 177  -3.0697531900  0.8661691201 -1.630985427
## 178  -2.7101614693 -1.2727964731  2.152018855
## 179   4.1285265136 -0.6137671561  0.134799647
## 180  -2.1237709382 -0.4695014769  1.527021506
## 181  -0.6136743525  1.2391646980  0.407291993
## 182  -3.9539294821 -0.0401647682  0.415809059
## 183   2.2085514439 -1.0074318943 -1.088857817
## 184   3.5781838719 -1.1264866712 -0.500212836
## 185   3.7415778051 -1.1455748228 -0.063675779
## 186  -1.0789945701  2.2919410914 -0.880672285
## 187   0.7039770325  0.9423306914 -1.200893505
## 188   3.6786906724 -1.7229238679  0.443811374
## 189   3.2002412419  0.9680757429  1.297290249
## 190  -3.8930979155  0.6999328325 -1.671344042
## 191  -3.0201074062 -1.6663540038  0.812074252
## 192  -2.8217172182 -1.1321253405  1.512781445
## 193  -0.0453037123  1.6470339743  0.380133521
## 194  -3.3167412283 -0.7206107820 -1.244958762
## 195   3.5823761834  2.9467907415  1.287755739
## 196  -0.1787307356  0.3616264148  0.190394831
## 197   3.1263691886 -1.6929677091 -1.478968556
## 198  -2.1230829091  4.1067254336 -0.423205036
## 199  -0.1630088796  3.9759948599  0.637544292
## 200   3.0887191041 -2.0534428030  0.747503464
## 201   0.4096798652  1.5378526643  1.398915552
## 202  -4.9429283408 -0.4711171237 -0.496374512
## 203   4.0930884560 -0.1452542326 -1.772725613
## 204  -1.2094556450  1.1268149563 -0.967550776
## 205   2.9307777616  1.2757881788  1.467995834
## 206  -1.4250737868 -2.3560162581  0.744935817
## 207   2.4877927894 -1.7222833212 -0.413724664
## 208   1.1024533527  0.8887591842 -2.673322444
## 209  -2.4896362944  1.3957844810 -0.802583299
## 210   2.9024348541  2.4494977086  1.003435788
## 211  -2.0313602438  0.9667268673  0.648755746
## 212   0.2250672387  1.5400363083  1.557902713
## 213   0.5712673475  1.9178066254  0.141166137
## 214   3.4960568403 -0.8179890014 -0.536330953
## 215  -4.5165264447 -1.2375724332 -1.280274789
## 216  -0.0444893781  3.1089027943  1.025643900
## 217   2.7477707609 -0.3735219668  1.039364073
## 218  -3.8041033292 -1.0748187850  0.416035870
## 219  -3.6760575904 -2.5209269373 -0.846152492
## 220   2.8082459059  1.3176599572 -1.016281905
## 221  -2.3564219884 -1.1041864798  1.768495451
## 222  -4.5026749752 -0.7048844433 -0.593799254
## 223   1.7063546635  0.1682042383 -2.047792132
## 224  -2.7996327272 -0.0920344034  1.600314317
## 225   4.2321746359 -1.1500531719  0.329236340
## 226   2.6897284103  0.0233739055  1.102959324
## 227  -4.1187611118 -0.6191424066  0.345801397
## 228   4.3534573460 -1.9382626323  0.675319934
## 229   1.5866400117  3.9178196860  0.799612634
## 230  -4.6317499686  0.0316170136 -2.429680562
## 231   2.4277530530  0.5174455324 -0.394109486
## 232   2.9160195968  0.8386991532  0.222869915
## 233   4.1550623247 -0.8675825177  0.725320642
## 234   3.1188321083  1.7321387578  1.193664811
## 235  -2.7343516708 -0.1689000751  0.296808149
## 236  -3.7008377204 -1.1513988867  0.856447630
## 237  -0.1822427364 -0.7003299903  2.169782103
## 238   2.3681177998  2.2128511938  1.278882175
## 239  -3.9309976486  0.1919668929 -0.974881945
## 240   1.5958715714  1.9732982999 -0.633066961
## 241   0.6551597038  0.1717245083  1.069044352
## 242  -3.1023667424  0.2073952435  0.747308508
## 243  -4.8460102401  0.9646253464 -2.031684899
## 244  -3.3644535448 -0.7812331414  1.544980702
## 245   1.4779686587  2.3036673766 -1.277902366
## 246  -0.5316587620  0.2336451341 -0.694131550
## 247  -3.8881441314  0.0189901043 -0.095719436
## 248  -1.0984735682  2.8768633507 -0.729934110
## 249  -2.2793552818  1.2518020181 -0.104310623
## 250   4.3237403333 -2.0704633107 -0.374767743
## 251  -1.7191420023  2.1927897453 -0.383285516
## 252   3.0870063787 -2.0552518000  0.699846559
## 253  -2.3950122314 -0.7261074308  0.750844542
## 254   3.0639820197  0.0910143404  1.128873429
## 255  -2.5247769419 -0.7581208787 -0.434982697
## 256  -1.0239811830  0.0333686602 -2.067019897
## 257  -2.9776227192  0.1493457989 -0.277265561
## 258  -2.2366818055  0.5147723985  0.393608249
## 259   1.5701048953  0.8284179767 -1.674624152
## 260   3.5766229988 -0.3193519552  0.936571224
## 261   2.2468887085 -2.6204431160 -0.549625231
## 262   2.6195851902 -1.4709983746 -1.181757758
## 263   1.9656284111  0.9486870018 -0.113828942
## 264  -1.9688589003 -0.9520899295  2.543016364
## 265   0.2250672387  1.5400363083  1.557902713
## 266   0.1459414193  0.1781602009  0.948443732
## 267   1.2146643497 -0.5925564104 -1.551965760
## 268  -3.8111942214 -0.1768763651 -1.741142097
## 269  -2.2994463731 -0.7438722120  0.907654271
## 270   3.7373501964 -0.7875206747  0.084429751
## 271   2.2471386157  4.3934121729  1.257881239
## 272   4.1456423356 -2.0516665795 -0.212954898
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## 585  -1.5104363012  1.5884727831  1.787726523
## 586  -3.1067010992 -1.0000362343 -0.903058769
## 587  -4.1144773028  0.0607348017 -0.319357643
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## 593  -4.3445165243  0.8168717587 -0.349659185
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## 595  -0.0559817308  1.6535746291  1.210190510
## 596  -4.4747146219  1.1409011039 -0.987677723
## 597  -2.6275904125 -1.3062288503  0.276145679
## 598   0.2617046409  0.7675586607  0.062805219
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## 600  -1.6556517725 -1.5128312348  1.267095171
## 601   2.7118462803 -1.6119634019 -1.514842541
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## 603   3.5066626323 -1.0414142230  0.401464558
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## 605   3.5003286628 -0.5407909611  0.927330495
## 606   3.7753295384  2.8546121032  1.187678083
## 607  -2.2836918857  1.2566999040  0.787192279
## 608  -1.6865330498  0.6946586579  1.606298189
## 609   2.7834426184  1.3260677193 -0.378532140
## 610   2.4002455721 -1.3434955683 -2.047597741
## 611   2.9785768003 -0.0746263486  1.465582742
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## 613   4.0073101873 -1.1768207605  0.226859216
## 614  -5.0452351918 -0.1184454688 -1.609730481
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## 616   1.0553680322 -0.6861065105  0.318482706
## 617   3.1316590398  2.5288487796  0.574970299
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## 619   3.2045586811  0.7813068317  0.895887749
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## 623   3.3915227840 -0.2676111368 -1.702068216
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## 628   4.3978033382 -1.8611449355 -1.987029002
## 629  -3.3495154427 -0.1399922722  1.558126770
## 630   3.0963401858  1.7914210562  1.462965973
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## 632  -0.3573540739  1.4915425048  0.401151732
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## 635   3.1000045879  1.2766144138 -1.707122879
## 636  -1.4641750876  2.1497551993 -0.988587317
## 637   0.2475165722  1.8924101783  0.232765735
## 638  -4.1314385475 -1.4651310876  0.948614455
## 639   3.2726188317 -0.4279702041  1.149627693
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## 646  -0.6083383719 -0.1295572705  0.743311433
## 647   2.9909083090  0.8604728929 -0.710221375
## 648   0.9751789497  2.0859833984  0.067109035
## 649   2.7119370961  0.2261764268  1.748618638
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## 651   0.4742783623  1.2808602107 -0.279661785
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## 653   2.5062151993  2.0410115409 -0.891268572
## 654   2.4946841297 -2.3963276885  0.048986939
## 655  -1.3831100431  0.4108809613  1.699607187
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## 657   2.3634324361  0.6864308006  0.384292139
## 658   4.7251483693  0.6563420727  1.897149223
## 659   0.8276446403 -0.9505123327  2.658173415
## 660  -0.4324660132  0.1314697669  0.031537503
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## 662   2.1517410637  0.2175552847 -0.296244099
## 663   4.7119209699 -0.6290261485 -0.363138329
## 664   3.0200889983 -1.4755108603 -2.224053685
## 665   2.7172644592 -1.0970578347 -0.973571592
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## 667   1.9353399306 -1.1602739770  0.283654303
## 668   0.1819619266 -0.2814042208 -0.208132851
## 669  -3.4597118884 -1.6313228189  0.665017024
## 670  -2.1598952179  1.3563048929 -1.332006714
## 671  -1.5393458905 -0.1704995171 -1.500382168
## 672  -3.6945965695 -2.5551080114  0.333102126
## 673  -3.1429458440  1.2603295200 -1.569005695
## 674   1.8685080772 -0.0965505928  0.820147948
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## 676   2.0395984044  0.5194493576  0.551061954
## 677   1.7536117917 -0.0257760695  0.143737096
## 678   1.6873557966  0.9996824802 -0.025941677
## 679   3.2509114403 -2.0549024589 -1.864659354
## 680  -0.9911057861  0.2369561490  0.107391889
## 681   3.6076561599 -0.1095651065  1.112323884
## 682  -2.4701280973 -1.4374322472 -2.011700853
## 683   0.5313171844  1.4688682942  0.187960813
## 684  -0.0708525551  2.0067041559 -2.193886697
## 685  -2.5949342469 -0.5345156182 -0.630888057
## 686  -4.2752355402 -1.4129974302  1.122490317
## 687   1.8947302734  1.8391554996  0.304976783
## 688  -1.6061412501 -1.0431182647  1.339941221
## 689   1.7670459936  0.1007257538 -0.137656898
## 690  -0.6161718996 -0.1231209252  1.034182806
## 691   3.2678999651 -1.0353363335  0.157345220
## 692   3.8008383531 -1.0851132599  0.824594986
## 693  -2.9117355613 -0.8125124565  0.941510385
## 694   2.7988796050  2.3224462687  1.362785872
## 695  -4.0307410800 -0.7353238424 -1.123450375
## 696   1.1351300145  0.8262269184 -0.873855689
## 697   1.8223625493  1.0375098007  0.320116479
## 698  -4.5240647883 -0.9128717217 -1.172180496
## 699   4.0676322326 -1.3037283678  1.044627680
## 700  -2.6729377616  1.1541977835  2.129077032
## 701  -3.3167412283 -0.7206107820 -1.244958762
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## 704  -0.4899983176  3.4378210417 -0.169248591
## 705   3.6442872415 -0.5399919613  0.896433782
## 706  -4.7674304660 -0.0927498149 -2.701533787
## 707  -0.2878583572  4.5649931740 -0.327398185
## 708   1.3023736277  1.7083868510  0.848313769
## 709  -1.1951510769  3.4369610224 -1.093253037
## 710   2.2850663277  0.5818255633 -0.140861567
## 711   3.3124689890 -1.1719849645 -3.116378521
## 712   3.2708461125 -0.1387393192  1.395603756
## 713   1.1343746765 -1.0104163801 -1.606139986
## 714  -4.6334677606 -1.9465675154  1.863312575
## 715   3.0375887997  0.4007863542  1.354639326
## 716   2.4261769688  1.6582697537 -1.699176027
## 717  -2.4178700583  1.0504027068 -0.434811792
## 718  -2.8727123357  1.1961525412 -2.537793716
## 719  -1.4887908262 -0.9140858399  1.875510128
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## 722  -3.0472751614 -1.1093159777  1.562186271
## 723   3.5307061441 -2.6792229623  0.235975355
## 724  -3.4967738520 -1.0671361232  0.230759769
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## 726   4.0499815590 -0.8490810300 -0.885250852
## 727   0.3815681165  0.2662545587 -1.905673833
## 728   3.2726188317 -0.4279702041  1.149627693
## 729   4.2585690868  0.0686396569  0.149442587
## 730   3.7412131896 -1.4384707415  0.387075035
## 731   3.4030129160 -1.0484265936  1.616191012
## 732   1.8599448696  2.1902409126 -0.292763760
## 733   0.8546716575  1.8428402832  0.397965242
## 734  -2.0902063678 -0.4862993601 -0.460826109
## 735  -2.6756961419 -0.0628826459 -2.486974799
## 736  -2.3306438506 -0.9195002855  1.644844707
## 737   1.2956374804  0.5782397003 -0.340909847
## 738   3.7133423333 -1.2340226123  0.413040196
## 739  -2.4200640501 -0.2061203889 -0.889594981
## 740   3.2613221146 -2.0174112934 -0.914328302
## 741  -1.0028525553  0.7641398448 -0.174155894
## 742   3.1426258613 -1.0288524138  0.594466777
## 743   4.2115862177  0.1053838359  2.258731860
## 744  -0.4849797017 -1.3105253954  1.190971364
## 745   0.2490536025  1.7675931041 -0.699521853
## 746   2.6660002759 -0.5903183994  0.606444497
## 747   3.3801080429  0.2988296856 -1.402336304
## 748   0.8678797763  3.1609891397  1.309659500
## 749  -2.7043651382  0.5630875958  0.400178429
## 750  -4.1052366368  2.0087792029  0.864439192
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## 754  -3.0517068191  0.0022166365 -1.753498535
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## 762  -3.4968086549  1.3018029887 -0.777502632
## 763   2.1805051439  1.8785866477  0.209578880
## 764   0.8321085969  1.7607172368  0.256848477
## 765  -0.1460326111  1.6723295139  1.973991855
## 766   3.3965949468 -1.1282661634 -0.702224025
## 767   3.1676969108 -0.3612811984 -1.668752362
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## 772   2.1148777618  1.4023899760 -1.341774660
## 773   2.5321567584  1.9016471482  2.228301674
## 774   3.9305142946 -1.1828196403  0.411668481
## 775  -2.9015497559  0.2957778137  0.709808204
## 776   1.3777086450  0.5755246029 -0.301712054
## 777  -3.0862365582 -0.3349867159 -0.705715862
## 778  -1.0214657499  0.4263356343  0.692289970
## 779   4.2888592820 -0.3717992156 -2.086125679
## 780   3.4933739492  0.0036869728  0.237156233
## 781  -1.3925978009  3.1242864023 -0.475363211
## 782  -2.3356007678 -1.8239300819 -2.090506726
## 783   0.9471594946  0.2174457817 -0.021215427
## 784   1.3767751685  2.3324152076  1.464665753
## 785   3.6871752768 -0.7372931601  0.522173789
## 786  -2.3409605830 -2.0680100796  1.344186465
## 787  -3.0309739745 -0.4106862910  0.588467510
## 788   2.5494796113 -2.1559648634 -0.073106759
## 789   2.1671475018  1.6920871752 -0.279692748
## 790   4.2829635199 -1.6711888647 -0.728744177
## 791   0.6904390407 -0.2061143715 -2.352701111
## 792  -4.2010488091 -1.1575723326 -0.287439090
## 793   0.7487707012  1.4393459189  2.086121725
## 794   1.0088800202  0.3994799649 -0.688665685
## 795  -2.7898676111 -0.9520030757  1.919835049
## 796   3.5914579683 -0.9101264017 -1.282695937
## 797  -2.7387974002 -0.7423194968  1.062082404
## 798   0.9270360403 -0.3504657251 -1.410268251
## 799  -1.1345865836 -0.8756485144  1.168513689
## 800   3.9362388918 -0.1011717480 -0.877971360
## 801   3.3700574486  0.6755053541  1.266056811
## 802   1.4636729771  1.8270624226 -1.506864475
## 803  -4.0240566141  0.7202528444 -0.277286367
## 804   3.5088497183  0.4076987617 -1.176012146
## 805  -4.4181821244 -1.9126405356 -0.093116019
## 806  -2.3341418702 -0.3725189774 -0.947176262
## 807   1.3745342913 -0.2525829966  1.925216497
## 808   0.1120011575  1.8895412881 -1.552048967
## 809  -3.2211834764 -1.0888631222  1.772109083
## 810  -3.8628019068  0.1842685721 -0.102629758
## 811   3.6531190326  0.8925595702  0.654846605
## 812  -0.2564873002  0.9070370281 -0.007598237
## 813   4.3561834070 -0.4154366069  1.214708723
## 814  -3.3597736182 -1.1378468808  0.462033880
## 815   1.4876162149 -0.0243929194 -0.074777687
## 816  -2.0530665573  0.2715250595  0.005046854
## 817   0.2471419135 -0.0194372710  1.734853315
## 818  -1.9093338872 -0.9568516760  1.237849585
## 819  -2.6919527621  0.4379868405  2.549897741
## 820   3.3909340682 -1.8897591959  0.245145111
## 821   4.6167147437 -1.6764509130 -0.727985468
## 822   2.6511678753 -0.9549311470  2.045211988
## 823   1.2437149587  1.9268530875 -0.434333794
## 824  -0.7855151833 -0.2251446048 -0.032553845
## 825  -3.1079866793  0.3453754984  1.090348765
## 826  -0.6599481472  0.3791680884  0.309308558
## 827   0.7506349783  0.7904782225 -2.606283209
## 828   4.9613839422 -0.6231907815  0.943111431
## 829  -3.9593527200  0.0565225600 -2.368272530
## 830  -0.9795250129  1.6863161901 -2.312290363
## 831   3.0263877277  2.1709289726  2.275580078
## 832  -4.1135793779 -0.8480200740 -1.883024878
## 833   1.7940896426  0.6493759230  1.429953049
## 834  -3.9549411663  0.7060055425  0.124679493
## 835   3.7000313637 -1.6021788738  0.512218341
## 836  -3.6764165149 -0.8290772167 -0.312068608
## 837  -1.6526988676  2.6164578685 -0.761987349
## 838  -4.3715025062 -2.4111661105  1.311252611
## 839   0.3002707148  0.6854878096  3.793967894
## 840   0.4845579675 -0.2522670676  2.177113426
## 841   0.4028095207  0.5720174415 -1.156846733
## 842   0.1384679196  0.5174178630 -0.441990530
## 843  -5.3758140903 -0.5997364014 -0.418008845
## 844   1.4437587579 -0.7374139296  1.203104711
## 845  -2.3350519699 -0.6459896876  2.021821555
## 846   1.2420370293 -0.5046826115 -2.704082314
## 847   2.4420762989 -0.7863372936 -0.435369640
## 848  -3.2331890105 -0.6250562172  0.435712436
## 849  -1.1095233011 -0.1116485763  1.482367889
## 850   2.8762423627 -0.3813394823 -0.024711396
## 851   4.2371060019 -1.3634391389 -0.689032022
## 852   0.4909149224  2.6020052614  0.501552159
## 853  -1.4696648534 -0.2692670117  1.346170352
## 854   4.6060578732 -1.3013754786 -1.640220065
## 855  -1.8732391888 -0.2205722882  2.678391260
## 856  -2.1079104267 -0.6014551193  0.680731041
## 857  -2.7459308086  0.2210314003  1.168271752
## 858  -4.4526480915 -0.9365841533 -0.463401722
## 859  -2.9782552261 -0.8607117929  0.241537321
## 860   4.7119209699 -0.6290261485 -0.363138329
## 861   4.2566581683 -1.0357849906 -0.796860663
## 862  -1.6971828172  3.6485685952 -0.733456972
## 863  -3.8458557547  0.6761691547 -0.808374811
## 864   0.9306342326  3.2216589215  1.587953530
## 865   2.9088992217 -0.8747859520  0.170181017
## 866  -2.5198677490  0.0237581850  0.730395877
## 867   0.0614086047  2.3403950050  0.690533282
## 868   3.8743888645 -2.2356028409 -0.029414841
## 869  -4.7832147363 -2.5008062204  1.134888882
## 870   3.0366948807  0.5024741609 -1.315797012
## 871   1.7306595214  1.7902528686 -0.263622390
## 872   2.8700429935 -0.2533317150  0.232510250
## 873   4.0614987850 -0.0221085319  0.120685405
## 874  -0.2636129051  0.6857675630 -0.804636857
## 875  -2.2434717264 -0.0497279434  2.140463708
## 876   3.2198154113 -1.2966103580 -0.034111480
## 877  -1.8427134482  0.3722366841 -0.409689735
## 878   2.6703790426 -1.5474988168  0.281829887
## 879   1.0641478927 -1.3901242280  0.356628814
## 880  -0.9141208729  2.0487746548 -0.059612685
## 881   3.8269545287 -2.2448482532 -0.075724402
## 882   3.5528284631 -1.0947341046 -0.878078575
## 883  -3.5387263831 -1.1291308922  0.202241509
## 884  -2.7338341907 -1.2087002424 -0.857185727
## 885  -0.5167769539 -1.6997623294 -1.151502953
## 886   3.8017925800 -1.2630032592 -1.547737318
## 887   3.0397899605 -2.0506333726  0.428772375
## 888   0.9275699273  3.9151168740 -1.275568577
## 889  -1.0646266932  0.1004742160  0.004335313
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## 892   2.2264773626 -0.9690843719 -1.079910385
## 893   2.3850390674 -0.3928969356 -2.663557218
## 894   0.8042400322 -0.7495261975  0.764870338
## 895   3.5798669851 -0.1342693473 -2.150080423
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## 897   3.4279339021 -2.3391251757  0.912865496
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## 899  -2.5105679089  0.3678683218 -2.277170313
## 900   4.1129225326 -0.4029674525  0.626871574
## 901   0.5497025021  0.7272796714  2.174421905
## 902   3.3325130940 -1.8592020017 -0.148864166
## 903   4.2211653464 -1.8028422031  1.421504707
## 904   3.2075393427 -2.4012773759  0.890223085
## 905   2.9775652892 -0.0613375296 -0.977952671
## 906   2.3874134196 -0.8951361492 -2.227403847
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## 908   1.2747133662  3.0085815352 -1.471565889
## 909  -3.6066035453 -1.4372298152  2.775223407
## 910   3.2613221146 -2.0174112934 -0.914328302
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## 912   2.6994056639 -0.7667567443  0.759489500
## 913   2.7646590680  2.4674506804 -0.880584345
## 914   2.3502159104  0.7016436540  0.788246800
## 915   1.6620440717  0.8457215902 -0.676203775
## 916   2.2688396248  2.4056297390  1.894149391
## 917   1.3878163319  1.1380717533 -0.940202210
## 918   3.3314063384 -2.1327100990  0.592210031
## 919   4.9862283727 -1.2286810126 -1.023792813
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## 923   0.5642029489 -0.3874576977  0.667001653
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## 925   1.5573242276  0.3342499671  2.014083426
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## 927  -2.7504531671  0.5481714104 -2.153297170
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## 959   3.2002412419  0.9680757429  1.297290249
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## 961   0.3486107686  0.4057482364 -1.377003696
## 962   3.7601231387 -0.9953373633 -0.914305327
## 963   4.9887592718 -0.0995617888  0.118206808
## 964   4.3323946280 -1.7266396879  0.968880746
## 965  -3.5052758304  0.3215733923 -2.627098755
## 966   4.1266570656 -1.7485814239 -2.118740239
## 967  -4.2135674369 -1.0336257675  1.292199439
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## 974   2.7229853458  0.5511816407  0.990397302
## 975  -0.1256330016  0.6867496730 -0.143194778
## 976   1.8578304992  1.3104274209 -1.414292273
## 977   3.3453441853 -1.4655546689 -1.557602638
## 978   1.4270994701  1.9068515910  1.697732428
## 979  -2.8859078068 -0.9375422560  1.739108665
## 980  -1.2963685880  2.4645296990 -1.046765392
## 981   1.7612061209  0.4784207601  0.590280418
## 982   2.8813572169  0.8236879533 -0.096667584
## 983   4.0759771216 -0.9141829089 -0.622136294
## 984  -3.0263204839  0.7210300219  0.223694730
## 985  -4.4687533596  0.0862408909 -1.790493867
## 986   3.6401353035  0.8591862347  0.962114843
## 987  -2.1471619925 -1.3407132458 -2.775426946
## 988  -1.0341727860  2.0570201918  1.716142319
## 989  -2.5837668500  0.5911354563  0.006527726
## 990  -3.6786472071  0.2630069039 -1.723539346
## 991   1.5329476147 -1.4533313368 -1.288046208
## 992   3.8452956354 -1.7690502509  0.500538082
## 993   0.0486530395 -1.6075189093 -0.331948349
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## 997  -3.6956810362  0.0449537237 -1.382487140
## 998   1.2988876444  4.7444432425 -0.868551206
## 999  -3.3967370523 -0.7542901640 -1.860836262
## 1000 -3.0327829915 -0.3161307901  1.936781678
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## 1002 -1.8855878092 -0.9100141830  0.661688058
## 1003 -2.6811497641 -1.1370737788  2.896846484
## 1004 -4.0261284456 -0.5557595580 -2.312457182
## 1005  3.2189388029  1.9307424947  1.854181221
## 1006  4.1524596955 -1.1811203092  0.249543567
## 1007  4.3309046935 -1.2115262799  0.318189325
## 1008  2.9284060818 -1.9465699955 -0.257782239
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## 1012 -2.5705402192  0.8668285160 -1.966096896
## 1013 -3.1199580716  1.0756217470 -0.938084275
## 1014 -2.2256275748 -0.9783597403 -1.937600404
## 1015  1.0486669646  1.4761182824  0.156649774
## 1016  1.6813017932 -0.3123975228 -0.604036577
## 1017 -3.4786184759  1.5325691839  0.081402029
## 1018 -3.1901969700  4.1331735326  4.796162985
## 1019  3.0106222261 -1.6952817146 -0.406909355
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## 1021 -2.5183875675  0.9455338358 -0.398277691
## 1022 -1.2640198126 -0.8504772764 -0.058559720
## 1023  3.6891210358  0.0065011544 -1.467647958
## 1024  0.6451106512  1.6957269519  3.046693985
## 1025  1.2011742793  1.7216433398  0.487199940
## 1026  3.8452956354 -1.7690502509  0.500538082
## 1027 -5.0074264124 -0.1272171741 -2.324474397
## 1028  3.6933531945  0.8244061940 -1.545428252
## 1029 -2.8211502433 -0.7179018599  1.116033626
## 1030 -3.6845294216 -0.6719266897 -0.061004661
## 1031  2.3677841467 -1.0396860943  0.434178212
## 1032 -1.1399195656  1.6989767830  1.257607367
## 1033 -1.3539439662  0.8584716409 -2.169491664
## 1034  3.3390556714 -0.1548749629  0.096864267
## 1035  2.9999422061 -2.3687060751  0.790820259
## 1036 -0.1671950460  5.3579740881 -0.785932002
## 1037  2.1835240055  1.6713011834 -0.588017309
## 1038  2.2249822436 -0.7394259970  0.487116725
## 1039 -3.0763759055 -0.3210220547 -1.657175133
## 1040  0.6942486822 -0.4573685698 -2.814370058
## 1041 -3.4693896937  0.6533384281  1.036380062
## 1042  4.8648894764 -2.5406410403 -0.128152952
## 1043  3.6967984616  2.5309418995  2.544872439
## 1044 -3.9110885952 -0.9395109782  0.676727795
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## 1048  1.9857008561  1.8011654097  0.942247246
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## 1060  4.2954323915 -0.9556412432 -0.448091500
## 1061  1.3860234794  2.1589569465  1.003172559
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## 1091  1.5459033249  0.7420248919  0.057343207
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## 1105  3.0190633583 -0.2565806884  0.023919326
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## 1109  4.2092021349  0.1183356982  0.978074921
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## 1179 -3.3644535448 -0.7812331414  1.544980702
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## 1192  1.6672112303 -0.7510454801  0.046216839
## 1193 -3.0815287037  2.9857073616 -0.795233569
## 1194  3.1414011800 -0.9328825799 -0.987585387
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## 1196 -2.7623389606 -0.5001824371  3.489086563
## 1197 -3.9085657063 -1.1069278257  0.841326404
## 1198  3.3744520862 -1.2581221614  0.562600912
## 1199 -2.8863060304  0.3346363530  0.442539840
## 1200 -2.4852861062  1.2441454073 -1.423380095
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## 1203 -2.3540578957 -0.6785859374  1.371803533
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## 1205 -3.5550399400  1.0533447877 -1.994315694
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## 1208 -2.0990583136 -1.0489835271 -0.039829911
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## 1215 -1.7779020933  1.1341290952 -0.452674309
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## 1218  3.7155254197 -1.5331111898 -0.732378963
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## 1221 -6.4212269873  3.5548470483  2.890671379
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## 1499  3.5476560010 -1.7614763571 -1.234460643
## 1500 -2.6478247241 -0.9656096008  0.621706548
## 1501  2.2771999740  1.7938998537  2.098972200
## 1502 -1.1533452301  2.2241175682  0.371971815
## 1503  1.6821385241  1.1778545764 -1.671826913
## 1504 -2.4734534071  2.0022890933  0.227518847
## 1505 -2.5306132869 -0.4580217301  3.139310681
## 1506 -0.8302403309  1.1931623019  3.354900149
## 1507 -4.4637066903 -1.0953961012  0.954856749
## 1508  3.3674726069 -1.1896741535 -1.397257884
## 1509 -0.0494467370  2.2033101684 -1.607586507
## 1510  2.3936510109  2.0175369718  0.108047596
## 1511 -2.6166916812 -0.9384637457  0.295211341
## 1512  2.6975763752  2.1219254549 -0.134050419
## 1513  3.5707062046 -1.0395732654  1.660672607
## 1514 -4.4695512927  0.5054373726 -2.651947521
## 1515  0.8922471629  2.1506994932  0.679581832
## 1516 -2.4661358234 -0.2143565583  0.095597834
## 1517 -2.1021335373 -2.1022640572  1.884090656
## 1518 -1.1226676361 -0.0874260079  1.158967894
## 1519  2.1015926084  0.7016916536  0.776398906
## 1520 -4.0679414570 -0.9815917885  0.584276772
## 1521 -1.4847372250 -1.2039787693 -2.082086068
## 1522  2.6149992392 -2.1992267214  0.040596173
## 1523  2.4332561355  0.2232344503  0.742403262
## 1524  3.0106222261 -1.6952817146 -0.406909355
## 1525  3.0963401858  1.7914210562  1.462965973
## 1526 -4.0048177502  0.2983712764 -0.665655704
## 1527 -1.0183066660  3.1529725647  0.235384733
## 1528 -4.0764095387 -1.2974701241  0.550840089
## 1529 -2.1871782395  1.1708243252  0.619104576
## 1530  3.1658438485  0.9052811622 -0.013271522
## 1531 -1.4696648534 -0.2692670117  1.346170352
## 1532 -1.3925978009  3.1242864023 -0.475363211
## 1533 -4.4149636947 -1.0955728295  1.249487301
## 1534  3.3782967118 -1.8151709734 -0.712732806
## 1535 -2.9397249448 -0.4516984894 -0.294501870
## 1536  0.3682249880  0.4595846124 -0.360905690
## 1537  1.9364007317  1.5243534417 -0.465191939
## 1538  2.7293093307 -0.4780658605 -1.146170578
## 1539 -0.0528403024  4.6023133341 -0.939412045
## 1540  1.2357074810  4.5736181504 -0.765203464
## 1541  4.3490029545  0.4297254486 -1.743673033
## 1542 -3.4786184759  1.5325691839  0.081402029
## 1543  4.6710322242 -2.2376514875 -0.567367797
## 1544 -3.1385620613 -0.9695712224  0.387720653
## 1545  0.3347626304  2.2631576117  1.986873769
## 1546 -1.9941647728  2.1959902723 -1.541229519
## 1547 -1.4480056777 -0.7473618232 -0.027928801
## 1548  2.5070996742 -0.8704237619 -4.067109282
## 1549 -1.4422926608 -1.4645637963  1.337677885
## 1550  1.1866148607  0.1723388318  0.406113156
## 1551 -4.8575252601 -0.1796498043 -1.601092870
## 1552  2.8144326231 -1.0351595780 -1.757776256
## 1553  2.8240855884 -1.3118702249 -0.659834145
## 1554 -2.1129778331 -1.0867681209 -1.261387173
## 1555  3.8512495022 -2.4383211012 -0.434255529
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## 1557 -0.8273513363  1.5605549894  0.390836054
## 1558  1.1482371864 -2.2620224695 -0.162030615
## 1559 -1.5267804729  1.2465195684  0.790546468
## 1560  4.1577938477 -1.0856370994  0.677356593
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## 1562  2.8398264297 -1.5586647177  0.242526101
## 1563 -3.2181265476 -0.9622663699  3.001086039
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## 1566 -0.1470825371  2.6379284773  0.045608397
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## 1568 -0.3803431005  0.9119712804  1.513060957
## 1569  3.2735571194 -0.2601403044  1.070250058
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## 1571  0.4191394472  1.8954772507 -0.157939420
## 1572  5.3520738583 -0.5904426489 -0.248285033
## 1573  2.8621083203 -0.3617460773 -1.345301803
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## 1577  2.4134800263  0.4988930339  0.153803247
## 1578  1.6104179526  1.2703271517  2.621539544
## 1579 -0.5534504942  1.2999233492 -0.912833897
## 1580  0.6942486822 -0.4573685698 -2.814370058
## 1581  3.4617071991 -1.2059709897  1.508197718
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## 1583  3.4271772332 -1.1131243968  0.912047936
## 1584 -0.8688646607  0.8091321404  1.154818452
## 1585 -3.8100331023 -0.0513699047  0.276722591
## 1586  2.8299421721 -0.2908171617  1.566318639
## 1587 -4.3718902347 -1.2566015326  0.371907035
## 1588  0.2719205394 -1.3560966150 -0.097660751
## 1589  0.0414962046 -1.2488516868 -0.777970846
## 1590  1.6283339940  3.9725611615  0.616994866
## 1591 -2.2172294148  0.4296601667 -1.893703955
## 1592 -2.6861775904 -0.1866717193  1.294549648
## 1593  3.4752605404 -1.5778752397  2.221775016
## 1594  0.1150663479  4.0624506828  1.279835696
## 1595 -0.4502298254  0.4468645304  0.389213531
## 1596 -1.9388085532  1.0200716103 -2.742610718
## 1597  1.8604616652  2.0814379398  1.382846757
## 1598  1.9465943320 -0.3636119247  0.154384912
## 1599 -3.0027758671 -0.3760580168  0.173771601
## 1600 -2.1877542682 -1.0429970189  1.430779367
## 1601  3.1000045879  1.2766144138 -1.707122879
## 1602 -0.8823502153  0.8303697790 -0.983423430
## 1603 -3.0167685054  0.4588660500 -3.231792322
## 1604 -4.7225940819 -0.9626452024  0.200258543
## 1605  0.5691920147  1.0907242176 -1.386950495
## 1606 -0.0586076697  3.9942675325 -1.611333934
## 1607  1.0188274644  1.1433839926  0.297312266
## 1608  3.6352382217 -1.2725290137 -0.555100644
## 1609 -3.4023960958 -0.4230472107 -1.068095928
## 1610 -3.4342495676 -0.9184879674 -0.058819942
## 1611 -3.5261305848 -0.7798241507 -1.301916808
## 1612  1.1792801254  2.3604581153 -1.956867290
## 1613  1.0616275984  3.5381880994  2.788820656
## 1614 -3.6299149037 -1.8375179393  1.341828645
## 1615  2.4369253143 -1.1015466813  0.679027088
## 1616 -3.3501017678 -1.7072096125 -1.281151353
## 1617 -0.2444234745  0.7902962093 -0.624330929
## 1618  2.5746800365 -1.5601697046 -0.190251225
## 1619  2.9449030028 -0.4393497394  0.791362562
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## 1621 -2.7547778705 -1.5462963152  0.485101414
## 1622  3.6550141595 -0.3943387733  0.488947815
## 1623 -2.0586983890  1.3803632740  1.057591088
## 1624 -3.7885598327  0.3809834458 -1.931895269
## 1625 -2.7407589885 -1.0398115349 -0.651477010
## 1626 -0.4626260659  1.2706847174 -0.537360629
## 1627  2.5373708210 -0.1816905878 -1.913780702
## 1628  1.6214147611  1.3071989394  2.070676883
## 1629  3.7024798533 -2.1151016814 -0.462631972
## 1630  0.6306624220  3.5279754474 -0.166608088
## 1631 -1.2648103045  1.5559845998  1.349588072
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## 1638  2.7834426184  1.3260677193 -0.378532140
## 1639  4.4768645940  1.3012865097 -0.910376734
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## 1642  3.8823659684 -0.3757030120  0.130231896
## 1643  3.5066626323 -1.0414142230  0.401464558
## 1644  4.4137139628  1.3916864899  1.657686364
## 1645 -1.7415107411 -2.3773645467  2.471167941
## 1646 -0.6881718839  0.5137713063  1.657943794
## 1647 -3.7384318423 -1.6081223121  1.923131975
## 1648 -1.0533436384  1.4015620299 -1.502290600
## 1649 -2.6541965045 -1.3109368021  0.048357568
## 1650  4.2520035653 -1.4606830308  0.662539387
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## 1652 -3.7021861780  1.8217819990 -1.299254907
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## 1654 -0.3609760849  0.4544809193 -0.368482099
## 1655 -3.7364048697 -0.7435676488 -1.155107492
## 1656  3.6099671218 -1.0366825973  0.282160467
## 1657  1.1920677479  2.1977262484 -0.062696423
## 1658  2.8020317938  1.0668793277  1.254335567
## 1659  3.9101037716 -2.1161835832  0.755552547
## 1660  0.5680134957  1.8055747735 -0.181449539
## 1661  3.0358727830 -0.6513032443  1.002566459
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## 1663 -3.6865410023 -1.4281655525  2.454401780
## 1664 -2.4289872399 -1.2263634795  2.319925682
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## 1667  2.0769082465 -1.6436549535 -1.362564825
## 1668 -4.2432500343  0.8378936751 -1.753295440
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## 1680  4.8581237357 -2.0256966504 -0.918505702
## 1681  3.5087865476  0.6254110060 -1.653555072
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## 1683  4.3885063486 -0.7764023061  0.499923050
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## 1685  0.5012142714  1.7069320883  3.185780453
## 1686  2.1373838834  0.2795157868  0.361276104
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## 1688 -1.0616795459  1.3971194023  1.075157084
## 1689  2.6385097276  1.7579685242  0.344935802
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## 1691  4.5323995616 -0.4126499685 -1.600632730
## 1692  4.8561938141 -1.8614168759  0.479393293
## 1693  4.4345771189 -1.0853363379 -2.087511894
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## 1696 -2.1340191471  2.6242746862 -0.560818412
## 1697 -2.7923709622 -1.2851288140  1.088678377
## 1698  2.6975763752  2.1219254549 -0.134050419
## 1699  3.8977273059  0.5412707054  0.238247186
## 1700 -3.3365819753  0.5697202846 -2.489219287
## 1701 -2.7233563842 -0.8765589489  1.384920072
## 1702 -3.1970433753  0.1503402029 -0.484949075
## 1703  4.3938673386 -2.0470282792 -0.488513691
## 1704  2.1424015438 -0.7810769822  0.659847929
## 1705  2.3306192110  1.2186402214 -0.567201747
## 1706  1.2956374804  0.5782397003 -0.340909847
## 1707  3.5871280774 -1.1979728260 -0.691046015
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## 1711 -0.1825030907  1.4774977017  1.431751444
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## 1713  3.9576917599 -0.1884396300 -0.336750870
## 1714  0.2243663287  2.2818484168 -1.338698915
## 1715  4.3734757756 -0.9259464165  0.301925691
## 1716 -3.2290319812 -1.3208360022  1.326093111
## 1717  3.3696008563 -0.1029516327 -1.206057267
## 1718  2.8205899085  0.4819781234  1.174733105
## 1719  3.9765282424 -0.3435076788 -1.887365788
## 1720  1.7397679513  1.9620931634 -0.772153428
## 1721 -0.0535973950  1.4430179566  1.716292069
## 1722 -3.0042399858 -2.1249124402  1.144403353
## 1723 -0.2291687211  2.6489206835 -1.852523360
## 1724 -3.2993823020  0.6631578075  1.002082459
## 1725 -3.1176881942 -0.9473655187  1.574681029
## 1726 -2.4570217146  0.4814364317  0.304997019
## 1727 -1.5325861533  0.6645022139 -0.089083910
## 1728 -3.2999070039  0.0697966422 -2.652268420
## 1729  2.4289727043  0.1702584499  1.086971206
## 1730  2.2576568716 -0.1059171762  1.644775455
## 1731 -3.3193223879 -0.8415456667  1.644301142
## 1732  1.3473478682 -0.8789825336  0.445248131
## 1733  3.0747810402  0.0875499577  0.581832070
## 1734 -2.7999331688 -1.4205629353  2.016811533
## 1735  2.9622286281  1.2401689510  0.695808780
## 1736 -4.2073814439 -1.2943858022 -2.073258514
## 1737  3.3987709470 -2.4361691641  0.596489527
## 1738 -2.7704369342 -2.2869429481  0.708948897
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## 1740 -2.0432118083  0.9752979927  0.449334659
## 1741 -0.3452093894  0.0631030607 -1.471900494
## 1742  2.3677841467 -1.0396860943  0.434178212
## 1743  3.7753360944 -1.4219709721 -1.730077422
## 1744 -4.2851414488 -1.2197494527 -1.523105877
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## 1746 -1.5335110931  0.7197432338  2.065867208
## 1747 -1.3402237044  0.9639876510 -0.069594547
## 1748 -2.8164204370  0.3527894326 -1.416789428
## 1749  2.2914689018 -2.5966005339  1.200465677
## 1750 -3.0581811481 -1.2071572105  2.669916280
## 1751 -2.1578090439  0.5205844435 -1.553621332
## 1752  3.2119585730  0.2763142760  1.341842311
## 1753  2.2013249093  1.4041457845  0.240109799
## 1754 -0.6802897109  1.7535125026  0.800863929
## 1755 -1.2308709060  2.1144236216  0.980233759
## 1756  0.1167488352  1.8683844333  1.110495008
## 1757 -2.1019525843 -0.7096221049  0.617857263
## 1758  2.0225970149  3.8161366218  3.854220838
## 1759  3.2665328169 -0.5877542262  0.473173948
## 1760 -2.0530665573  0.2715250595  0.005046854
## 1761 -3.4472494557  0.0285259145 -3.063335794
## 1762  4.4401947386 -1.5452494896  0.041832211
## 1763 -2.0291800083 -1.4750013591  0.056937024
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## 1773 -0.0444893781  3.1089027943  1.025643900
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## 1775  1.7419336485  1.6932412686 -0.072024427
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## 1777 -3.1023667424  0.2073952435  0.747308508
## 1778  3.6498893076 -1.9593035861 -1.381976779
## 1779  3.4205988142 -0.6875049063 -0.168527319
## 1780 -3.9549411663  0.7060055425  0.124679493
## 1781  1.1544176803  2.9363427528  0.578111599
## 1782  2.3188894195 -1.6470516148 -1.669500431
## 1783  2.7490609010 -1.6068187093  1.959320453
## 1784  3.5247157418 -0.0861608496  0.350722066
## 1785 -4.5136320753 -0.1833126666 -0.820988988
## 1786  2.3530864355 -0.0957490207  1.721398473
## 1787  3.7537756402 -0.2900873745  0.398769424
## 1788  0.6305293379  1.3861835171  0.029224511
## 1789 -5.1147961957 -0.2495247710 -2.460927974
## 1790  2.0780594521 -0.1586682189 -0.100614545
## 1791 -3.5872033575  0.6494535930 -1.129561928
## 1792 -2.1489226976  0.6973931276 -0.339743103
## 1793  2.9828913057 -0.1955742765  0.650862230
## 1794  3.2212065055  0.2707700769  3.074438597
## 1795 -3.7353747220 -0.5194968460  0.148490087
## 1796  0.1881430008  0.7286022534 -1.013151232
## 1797  3.4508140955 -1.7947533193  0.732780963
## 1798  3.8235859179 -0.8716075634 -1.477869721
## 1799 -1.1893185967  0.2450236536  1.372831233
## 1800 -2.1396167077 -0.3302531381  1.563105761
## 1801 -1.6457357742  1.0175460897 -0.284003469
## 1802  2.0995960209  2.9132938847 -0.308047555
## 1803 -0.8489064444 -0.0240774067  0.255497221
## 1804 -0.6767270781  0.2024566964  0.361169508
## 1805 -2.5240344628  0.4874017102  0.489257642
## 1806  4.1376048285 -1.5214516977 -1.165350357
## 1807 -0.7036754008  0.8384882775 -0.437442445
## 1808  0.7971379574  1.0925662232  1.142638174
## 1809  3.3157600747  4.4400513684  0.722446465
## 1810 -1.9972175831  0.9537760748 -0.105220709
## 1811  4.1411056016 -1.4905535922 -1.358629943
## 1812 -3.4356840252 -0.7950471287 -0.964186518
## 1813  2.9395631524 -1.0169201687 -0.438901362
## 1814 -0.0009284632  2.1848089456 -0.387922135
## 1815  2.9292371286  0.1042052632 -1.337406910
## 1816 -6.8295483456  5.2541656033  0.914713136
## 1817 -2.3677740139 -1.9360108271  1.599834167
## 1818 -0.0836109629  1.4384110939 -1.796251933
## 1819 -3.6323547902 -1.5181369498  1.004010068
## 1820  4.4507288421 -0.9977855100  1.867176640
## 1821 -4.5586570858 -1.1702580912  0.620456982
## 1822  3.5253477185 -0.6755448131 -0.200203459
## 1823  2.3050846891 -0.9107022586 -0.611613356
## 1824  3.4210403939 -0.5016256152 -0.523616560
## 1825  1.8242036198  2.2218007322 -1.464171278
## 1826 -2.1971618159  1.3123630084 -0.806559313
## 1827 -2.1813333002 -0.9469979069  0.381322339
## 1828  3.9065176023 -1.3639673092 -1.188239287
## 1829  1.2772779710  0.8000935043  2.165398425
## 1830  1.7933043961  0.2689325026 -0.747821089
## 1831 -4.6163880763 -2.3966106119  1.374670959
## 1832  1.3222307112 -1.0779911155  0.822742099
## 1833  2.5909413951 -1.0957768283  0.569550838
## 1834  3.6305587229 -0.6236962189  1.091401644
## 1835  0.9044863992  4.4319408913  1.503479215
## 1836 -2.9707840605  0.4992383431 -0.581885284
## 1837 -3.9503002154 -1.9268869356 -0.161576325
## 1838 -1.1714982866  2.7426029501  0.923186584
## 1839  5.2612354365 -2.1795872162  0.158967042
## 1840 -3.0794794627 -0.5106367165 -1.380571406
## 1841  3.6153708784 -0.1326167688  0.828440038
## 1842 -3.4949310572 -0.2006208492 -0.775351347
## 1843  0.9640350298  2.7708783198 -1.601888598
## 1844 -2.7138029067 -2.0576660305 -1.692317576
## 1845 -3.0972394599 -2.7783524587  1.594057571
## 1846  2.3411287130  1.3194595161  2.017117220
## 1847  4.4374639348 -2.0496324221 -0.356209063
## 1848  2.8976076704  1.1003675237 -1.694415157
## 1849  3.0362256513  0.7598874520  1.142877211
## 1850  3.2877104278 -0.2984176510  0.376874586
## 1851 -3.4957750098 -0.3113573064 -1.956693518
## 1852 -2.4376451947  0.4665734357 -2.215932535
## 1853  2.9733424483 -0.3397025661 -0.619992205
## 1854 -3.1351828813 -2.5816398580 -1.590792335
## 1855  0.7807817289  0.3814828999  1.901666298
## 1856 -3.2720907928 -0.2049398036 -2.727829282
## 1857  3.9577273459 -1.1379336108 -1.121032887
## 1858  2.4981092965  1.1993207021  0.975247753
## 1859  0.3882511111 -0.0849604987  1.037429554
## 1860  4.1550623247 -0.8675825177  0.725320642
## 1861  4.9150795570 -2.1170065246 -0.593117029
## 1862 -2.8064842947 -0.5978638520  1.064792790
## 1863  0.4287386632  3.0661832401 -0.513377325
## 1864  2.2319296121 -2.5736576869 -0.022523500
## 1865 -0.1769074392 -0.0446190763 -0.045545019
## 1866 -2.7101614693 -1.2727964731  2.152018855
## 1867  3.7643681544 -1.2710182195 -2.297780932
## 1868  1.2183822381  0.2527309653 -1.894095219
## 1869 -3.0697531900  0.8661691201 -1.630985427
## 1870 -3.0440013820 -1.9838019187  0.326011913
## 1871  1.0050044327 -1.8459649115 -1.088364264
## 1872 -3.2529712407 -0.2677802049  0.248536345
## 1873 -1.4586305950 -0.0983055294 -1.757423284
## 1874  2.8750201549 -1.6210774280 -0.023145731
## 1875  2.7493953684  1.2195517310 -0.522783778
## 1876 -3.0854574339 -0.9364589506 -0.102517132
## 1877 -2.6579451881 -1.3604061950 -0.178329293
## 1878 -3.1136375507 -0.6112721789 -0.133349162
## 1879  4.1119173873 -2.6161649285 -0.672548536
## 1880  3.3681174110  0.4202549274  1.264252811
## 1881  4.1212805021 -0.8628994499  0.017047256
## 1882  3.2865226748  1.1701706535  1.287451713
## 1883  0.7836735474  3.9263220105 -1.136482109
## 1884  0.0516814572  2.8991163177  0.468111480
## 1885 -2.4728729850 -1.7515693534  1.157444726
## 1886 -3.1292352366  0.0014754093 -0.830057184
## 1887 -1.2038744781 -1.7197464092  1.554740510
## 1888 -4.4895852933  0.6737511430 -1.159594620
## 1889 -2.2289287084 -2.2443762008  0.437564238
## 1890  3.0647440016 -0.9835001000  1.181535870
## 1891  4.5807004112 -0.6207366070 -0.538393500
## 1892  4.3242298800 -0.0861219002 -1.348310953
## 1893  3.4166375454  1.5968114223  0.223595629
## 1894 -3.7241848331 -1.3243515170 -0.213020605
## 1895  0.0509662855  2.5487011933 -1.346185274
## 1896  2.7032910950 -1.6051203717  0.171813424
## 1897  3.4706476797  0.1634953346  1.124092749
## 1898  1.6058338475  2.5710468173  0.420923444
## 1899 -2.2517069596 -0.0272852325 -1.688433070
## 1900  1.3171611324  0.4439228445 -1.499501512
## 1901 -3.0632625680 -0.6367429085 -0.092355722
## 1902 -2.8143456064  0.0840040234 -0.576939591
## 1903  3.3764085466 -0.0306392464  0.072616397
## 1904 -3.3396043262 -0.2307682500  1.609862932
## 1905  1.2947318968  0.9888499195 -1.163388195
## 1906  0.3752918863  2.7811598439 -1.468731309
## 1907 -4.2472767521 -1.3221922405 -2.032528280
## 1908 -3.0584035858 -1.8517518898  0.527424873
## 1909  3.2994196573 -0.9690170353 -2.482542970
## 1910 -3.2813766268 -0.3209041311 -0.312410913
## 1911 -3.4183444809 -0.6074153330  1.056207021
## 1912  4.0154998649 -2.3072727389  0.576741342
## 1913 -1.3768955815  0.3302558806  0.944354460
## 1914  3.1970342787 -0.6476252209 -0.009657616
## 1915  3.6542305823 -1.4443549542  1.613813284
## 1916  3.1382774056 -1.5917343860 -1.275326153
## 1917 -2.6748575698  0.4734686876  0.578705068
## 1918 -1.4098810496  0.0501085768  0.124980729
## 1919  1.7753203853 -0.7800285375  0.665180485
## 1920  3.6419307793 -1.4874018464 -1.014620995
## 1921  2.7496033074  0.8627846934 -0.558777866
## 1922  0.8671907957 -0.3691499977 -1.678976802
## 1923  3.1579787510 -1.7548361620 -2.745839383
## 1924 -4.0884251944 -1.6148955408 -0.568333360
## 1925  0.7064417606  1.5023379522  1.241612303
## 1926  4.5545473161 -1.6150416387 -0.196651013
## 1927 -1.2598392589  0.4702164947 -2.288382322
## 1928  3.2003677929  0.8242864426  0.336376030
## 1929  1.8931337147  0.7383234425 -2.286445770
## 1930 -3.0319657798 -0.7750635164 -0.880050961
## 1931 -1.3753432663  1.1814377239  0.411003526
## 1932 -2.1292646902 -0.3682146373 -0.810540688
## 1933  4.5341592769 -0.6662962371 -0.198589876
## 1934  2.5519188361  0.3325536545  1.055930550
## 1935  3.6072838999 -0.5551210935 -0.682974159
## 1936  4.3649682963 -1.4945668582 -0.562033928
## 1937  3.3828064783 -1.7597647273 -1.972509548
## 1938  1.3436880825  0.2005020711 -0.196427345
## 1939 -3.2470967570 -1.5929020643 -0.877210317
## 1940 -0.6802897109  1.7535125026  0.800863929
## 1941  0.3191537234  0.3490851335  0.979026342
## 1942 -0.6083383719 -0.1295572705  0.743311433
## 1943  0.0405416493  1.4001380054  1.045623611
## 1944  1.7481421306 -0.8351587012  2.396969848
## 1945 -2.9305541141  1.6860115339 -1.225124192
## 1946 -2.9131900120 -0.8951304210 -0.293852924
## 1947 -3.6022471718 -0.5338691757  0.540835858
## 1948 -1.1758393687  0.1161849345  2.285010197
## 1949 -2.0432118083  0.9752979927  0.449334659
## 1950  5.3104411978 -1.8267962762 -1.256616170
## 1951 -1.0773361363 -0.5045980898 -1.133586533
## 1952 -0.3390636890  2.7637226541  0.932403310
## 1953 -3.8438867784 -0.7304714598 -0.058717800
## 1954 -1.3526934633  2.0858515695  0.586759928
## 1955  0.7969833930 -1.7947896985  1.287754222
## 1956 -1.7321651328 -0.0129419567  1.697923443
## 1957  1.0023189549 -0.6333670367 -1.101963500
## 1958  3.9189670889 -0.5644542145 -1.367887459
## 1959 -0.9539085593 -0.0557035824  0.410235632
## 1960  4.9732094590 -0.0946017603 -1.744139948
## 1961  3.2609783215 -0.7429163483 -0.999633622
## 1962 -4.1954613959  0.1635263592  0.327472988
## 1963  3.9657031599 -2.2402787113 -0.423757120
## 1964  2.8625624423 -0.7933022045  0.904183314
## 1965  1.4154131713  0.6049935386  0.156412087
## 1966 -0.5455817683  2.0766618205 -0.587298272
## 1967  0.9732740370 -1.8659765344  0.947344576
## 1968 -3.6064284057 -1.2619105588  0.248883708
## 1969 -4.4076225927 -0.9207609627 -1.470145263
## 1970 -1.4419271609  0.1383229965  1.978512356
## 1971 -3.9454653050 -2.5787870265 -1.471731065
## 1972 -3.1199580716  1.0756217470 -0.938084275
## 1973 -3.6177020480 -0.1467822659  0.223142953
## 1974  3.3798884295  0.8747000587  1.851543873
## 1975 -0.7032407410  1.5997399838  0.135089779
## 1976  0.3590716832 -0.3623444885 -1.865367800
## 1977  1.6114558928  1.1325107821 -1.473149824
## 1978  3.6252446705 -1.0498662639  0.283013767
## 1979 -1.5859784756  0.1182343689  1.174285146
## 1980  4.9478742757 -1.2271300368 -0.484758169
## 1981  2.1485595223  1.1340967040  3.555327941
## 1982  4.0453048721 -0.5837022285  0.339649855
## 1983 -4.0679414570 -0.9815917885  0.584276772
## 1984 -2.2573320776 -1.6916469056  1.403619094
## 1985 -3.6966846640  0.5371956795  0.368570739
## 1986 -4.6681893615 -0.1438608207 -1.514130828
## 1987 -4.5896885104  0.5509631623  0.651452242
## 1988 -2.1891199587 -0.8980762848  0.534834295
## 1989 -1.9842872996 -0.9861578286  0.313634345
## 1990 -0.4592721285  1.1661774250 -0.123657052
## 1991  1.8027436932  0.5957417339 -2.006107901
## 1992  3.5080524695  1.1270310913 -0.785559576
## 1993 -2.7596843082 -1.1909539027 -0.669303992
## 1994 -4.0586585052 -0.5594121133 -0.230160539
## 1995 -2.5746909688 -1.5557870164 -0.277899442
## 1996  2.2993824243  2.6731962458  0.167960881
## 1997 -2.3560131888 -0.0063780275  3.739093241
## 1998  2.4250421925  1.8891432581 -1.925705100
## 1999  2.4192049865  1.0766662496  0.222710698
## 2000 -3.6182304287  0.0449210830  0.604682407
## 2001  3.3869025957  1.4243079266  1.964156786
## 2002  0.3561694632 -0.1418959569  1.766572504
## 2003  1.2139465571 -0.4473927237 -0.508378256
## 2004  3.1635289523  2.1282442153  1.921475304
## 2005  2.0527780457  0.3536513595 -0.159168676
## 2006 -2.4626274216 -0.9911202405  0.976820745
## 2007  3.8352301160 -0.3224013181  1.483374977
## 2008  2.2957409731 -1.4977262072  1.769973036
## 2009  2.6437306069  3.8551460036  2.106376678
## 2010 -2.9382759370 -1.1859247508  0.822155134
## 2011 -3.1561029804 -0.7224873829 -0.838236251
## 2012  0.8404390314  2.0750058200  2.724400163
## 2013 -0.5570280907  0.0818227391  0.983961335
## 2014 -1.9533382512 -0.5655338729 -1.811020119
## 2015  3.4868059688 -0.5101648409 -0.216288067
## 2016  2.9720696709 -0.2264194591  1.386681017
## 2017  3.7933626082 -1.2981909522 -0.340824668
## 2018 -3.2617861773 -0.3016108674 -1.040934900
## 2019  0.1672995947  2.4670124477  0.790751358
## 2020  2.3215344924 -0.7546469418 -0.581791657
## 2021 -3.5381366346  0.8670057814 -1.858791104
## 2022 -2.9916057161 -0.5352539588 -1.529032244
## 2023 -1.4372201300 -0.9637419096  1.225861538
## 2024  2.7042117305  0.8585133327  0.714333114
## 2025  3.1658438485  0.9052811622 -0.013271522
## 2026  1.7725498070  1.9820013433 -0.244074240
## 2027 -1.6630077335  0.1405420536  0.661940635
## 2028  3.2390065310 -1.1394687250 -0.280414060
## 2029 -3.7351314157 -0.3339873619 -0.666701640
## 2030  3.0963401858  1.7914210562  1.462965973
## 2031  2.4675256396 -0.4206736673  0.028012278
## 2032 -1.1248908459 -1.4613809791 -0.376922822
## 2033  2.2846849426  1.3007559564 -0.881057926
## 2034 -1.8440019464 -1.2178167723  0.698558184
## 2035 -3.0582100179 -0.9317868901  0.598487419
## 2036  0.7356651889  1.0677809199 -0.420218020
## 2037  2.7606037280 -0.5380265396  0.195958561
## 2038 -3.7353747220 -0.5194968460  0.148490087
## 2039  2.2249822436 -0.7394259970  0.487116725
## 2040  3.6339925230 -0.7710389773  1.566864828
## 2041  2.0958909110 -0.7870435950 -0.172917877
## 2042 -4.0373919246 -0.7137701075 -0.608560093
## 2043 -1.7524287175 -2.2990538366  0.867781792
## 2044 -1.1967628766 -0.0950210975  2.333427017
## 2045  0.0605008967  1.8892853777  1.232388848
## 2046  3.8746042999 -0.4807630394  0.878102472
## 2047  2.6131926577  0.8320261836  0.874325247
## 2048  2.9647932664 -1.7124987505  1.742066859
## 2049  3.9815543981 -1.0060256029 -1.011183166
## 2050 -2.4608911646 -1.3097117821  0.240706330
## 2051 -3.6095503030 -1.3090147276 -0.850415445
## 2052 -0.1768499645  2.5314004550 -0.059367190
## 2053 -3.8846584767 -2.0946501112  1.321707858
## 2054  1.0398632459 -1.4754105592 -0.354409543
## 2055  4.6059600898 -1.9302713462  1.376796756
## 2056  2.2993824243  2.6731962458  0.167960881
## 2057 -3.3845458123  0.2572267202 -2.457671992
## 2058  4.1696915335 -2.1818030924  0.796019667
## 2059 -0.7844392457  1.7304389800  0.088646981
## 2060  3.0613815566  1.0957155352 -0.056040305
## 2061  1.6917146059 -0.0218863853  0.667126495
## 2062 -3.3230281078 -0.7888119277 -1.768013150
## 2063  1.8665825190  0.6420421797 -3.237942789
## 2064 -3.1314515516 -0.4718612058  2.477819559
## 2065  2.6961947215  0.2333340877 -1.820429915
## 2066 -2.2103753386 -2.0211764105  1.980071868
## 2067 -2.8708198523 -0.9361309214  2.758636956
## 2068 -1.0773361363 -0.5045980898 -1.133586533
## 2069  1.1810094016  1.0747371960  0.142936003
## 2070 -0.8882242749  1.4612282896 -0.846736577
## 2071  4.2572922024 -0.6825584001  0.035518050
## 2072  1.3618546797  2.4926628495  0.974657214
## 2073  2.2918373642  0.7769201338  0.152327918
## 2074 -3.8317262543 -0.6669234990 -1.408246666
## 2075 -1.4083294215 -0.7586255636  0.031298500
## 2076  0.1980664435  2.1790158091 -0.076243126
## 2077  2.6624948828  0.4618965146 -0.505759722
## 2078 -3.1389859494 -1.4064670506  2.098173806
## 2079  2.9031176515  0.2400879654 -2.637225851
## 2080  4.0408379702 -1.1442230682 -0.505978793
## 2081 -3.5712448251 -0.8679823953  1.040688984
## 2082 -1.6340832196  0.1501648798  0.689802223
## 2083 -4.9407507154 -0.6798981861  0.708531579
## 2084 -2.2657621986  0.8009262844  1.123766851
## 2085  2.3598621942  0.9775996265  2.251488898
## 2086 -3.2526131695 -0.0473904215 -2.006981792
## 2087  4.3920196220 -1.0244969653 -0.421082144
## 2088  0.8762051640  0.8285550249  0.067409608
## 2089  2.9311936525 -0.8350855736 -1.568690419
## 2090  2.1369111736  1.7820791652 -0.692463107
## 2091 -5.3944164651 -0.0304566486 -0.253924661
## 2092 -3.3322778244 -1.0753349790  0.335755918
## 2093  2.4487356516 -0.9178063305 -1.705622024
## 2094 -2.7790581522 -0.5265189314  3.362565207
## 2095  2.5518031739 -0.8094147577 -1.014510722
## 2096 -0.0657409406  0.1438592858 -0.627532687
## 2097 -0.9457925630  1.3340827983  0.295866377
## 2098 -3.8271377451 -1.6126324226  0.076403899
## 2099 -0.3384519833 -1.2708317703 -2.104563788
## 2100 -1.7947591065  2.0554681211  0.741773483
## 2101 -4.2187476983 -1.5590022352  0.033816227
## 2102 -3.3559656836 -0.4242544674 -0.580679629
## 2103 -2.2468476676  0.9189708544  0.450774834
## 2104 -2.0909741871 -1.7899515172  0.070108691
## 2105  1.3412981049  2.1136197208 -0.341427372
## 2106  3.5264386626 -0.0702603205 -0.019814419
## 2107 -4.0116110961 -0.1790518272 -1.669456863
## 2108 -2.9161073653  1.2013208167 -1.479691294
## 2109 -0.1162850550  1.1829292567 -0.944258607
## 2110 -2.7191689899 -0.1937162279 -2.353562152
## 2111 -3.3818301655  0.7067047165 -1.508932218
## 2112  3.4072066995 -0.3869115490  1.382952898
## 2113 -2.5064884971 -0.3612021285  1.236951713
## 2114 -0.9457925630  1.3340827983  0.295866377
## 2115 -3.4342495676 -0.9184879674 -0.058819942
## 2116 -2.6079593656 -1.4531225222  0.614623739
## 2117 -1.2116634573  0.5417300652  0.739901845
## 2118 -3.2611160859 -1.8825101881 -1.381160562
## 2119  1.2470164107  0.9279206369 -2.354764344
## 2120  0.2807293675 -1.3593980587 -0.433171885
## 2121  3.5701208783  1.8681675940  0.195417423
## 2122 -0.1858052043 -0.6282026909 -0.153357183
## 2123  0.6623791364  1.4319496406  0.674866110
## 2124  4.3611727192 -0.9430641356  0.436531744
## 2125 -3.9809990447 -1.5956126339  2.177734520
## 2126 -2.7362033218  0.3636979084 -0.741150225
## 2127  3.5247157418 -0.0861608496  0.350722066
## 2128  2.8451143758 -1.5082114741 -0.997984426
## 2129  4.5899844734 -2.1035976399  0.605116736
## 2130 -2.9444056802 -0.8539024025  0.114845368
## 2131  0.7210673944  0.2635453652 -0.928830914
## 2132  2.8260001231  0.7444412963  0.287217204
## 2133  2.8902941581 -0.8480313767  0.673836234
## 2134  2.6289146531  0.5888451803  0.103250601
## 2135  0.7494188614  1.2573430706 -1.819548770
## 2136  4.5190491949 -0.9644995624 -1.050377087
## 2137  4.0175842422 -0.7676011911  0.434280807
## 2138 -0.2909789169  2.6491336138  0.184694865
## 2139  0.8491698954  0.5674631016  0.983262016
## 2140 -0.4324660132  0.1314697669  0.031537503
## 2141 -3.1166121336  0.2963181770  0.695119855
## 2142 -1.1500469112  2.5082861000 -1.537969192
## 2143 -0.8396558548 -2.5623084667 -0.122931390
## 2144 -2.2662348083 -1.3153401493  1.984971128
## 2145 -3.3413324947 -1.5777610889 -1.506615114
## 2146  3.1056292567  1.1343360552  0.254150071
## 2147  2.5288245167 -0.1470253890 -1.425664537
## 2148  4.2733878410 -0.8949644066 -2.312409147
## 2149  2.5335605858 -0.7455788020 -1.039761135
## 2150 -2.6676372477 -2.1275022762  0.462763817
## 2151  4.4004113717 -1.1891415655 -0.353472716
## 2152 -3.1891774979 -0.2158785772  0.178867912
## 2153 -3.8651115309 -1.1640551994 -0.363254338
## 2154  5.1052803221 -0.6343959179  0.804024963
## 2155  3.5668557775  1.3231393475  0.872018112
## 2156 -3.8736367474  1.4179556425 -0.353717473
## 2157 -1.2806142104 -1.4207454827  0.075762028
## 2158  0.9636916126  0.3760927684  0.433707840
## 2159 -0.1985713593  2.9153096636 -0.299026613
## 2160 -3.4900028735  0.7224561138 -0.252612927
## 2161 -2.2191396779 -1.2591363171  2.875547377
## 2162  2.1557558117  1.3477489725  1.272589705
## 2163 -0.1470825371  2.6379284773  0.045608397
## 2164  3.1860940921 -0.7738781483  0.756127584
## 2165 -2.4728729850 -1.7515693534  1.157444726
## 2166 -2.4797783193  2.7377547579  0.764269848
## 2167  2.3316259632  1.1437132806  2.385811682
## 2168 -3.6213863820  0.9501445875 -1.353709806
## 2169 -3.2189504164 -1.0630816194  1.838165591
## 2170 -3.3459407591 -0.0268111910 -0.845443352
## 2171  0.3485281893  2.1897938102  1.268866025
## 2172  3.5124472934 -1.6517066124 -0.061004172
## 2173 -3.5441245627  0.1662697310 -2.049992854
## 2174  3.4853422475 -1.8200980230  1.932245663
## 2175 -4.5880622878 -0.5710562165  1.313319103
## 2176 -4.2236372715  0.0854627696 -0.063214512
## 2177 -3.0314123755 -0.5360920966  1.008749274
## 2178  3.9873138038 -1.8513414954 -1.967074898
## 2179 -4.7225940819 -0.9626452024  0.200258543
## 2180  0.2691707977  1.3922542308  1.690820211
## 2181  1.1204562040 -0.2943990285  0.738033312
## 2182  3.9284316382 -1.4812804902 -0.890546150
## 2183 -2.9875007297 -0.4701000687  2.281780191
## 2184 -3.2012335538  0.4050796181 -1.523044123
## 2185  0.8654217679  0.8101318167 -2.209580159
## 2186  0.5909890805  3.0335309168  1.477352884
## 2187  0.7637774066  3.5558811704  1.622757106
## 2188  0.3169692393  1.3756157200  0.011940321
## 2189 -3.5404799578 -0.5562583787  0.187346988
## 2190 -2.6682146310  0.0839695260  1.581629015
## 2191  1.9041907564  1.4121932007  0.602560833
## 2192 -3.3564723314 -2.1711185516  1.109076641
## 2193 -3.4454583459 -1.2406639968  0.743889507
## 2194  2.6559170909  0.5366259407  1.175465353
## 2195 -1.3407201927  4.2745295166 -2.960382019
## 2196  2.2098320655 -1.3806840407  0.045170615
## 2197  2.7966772976 -0.0771899288 -1.270260193
## 2198 -1.3785120728  2.0657377203 -1.119772435

Clustering

Elbow/Silhouette methods for the selection of optimal number of clusters

# Elbow method
fviz_nbclust(Customers_clust, hcut, method = "wss") +
geom_vline(xintercept = 4, linetype = 2) +
labs(subtitle = "Elbow method")

# Silhouette method
fviz_nbclust(Customers_clust, hcut, method = "silhouette") + labs(subtitle = "Silhouette method")

Hierarchical Clustering (bottom-up agglomerative approach)

• Step 1: preparing the data

• Step 2: computing similarity information between every pair of objects in the data set

• Step 3: using linkage function to group objects into hierarchical cluster tree

• Step 4: determining where to cut the dendrogram into clusters

hc.complete = hclust(dist(Customers_clust), method = "complete")
hc_complete = cutree(hc.complete, 3)
table(hc_complete)

## hc_complete
##   1   2   3 
## 996 894 308

Dendrogram plot

fviz_dend(hc.complete, k = 3,  # Cut in three groups
  cex = 0.5,  # Label size
  k_colors = c("blue", "red", "green"),
  color_labels_by_k = TRUE, # Colour labels by groups
  rect = TRUE, # Add rectangle around groups
  ylim = c(3,15)) # Zoom in the dendrogram

## Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
## of ggplot2 3.3.4.
## ℹ The deprecated feature was likely used in the factoextra package.
##   Please report the issue at <https://github.com/kassambara/factoextra/issues>.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

K-Means Clustering

We do the k-means clustering to compare the clustering results we got from hierarchical clustering.

set.seed(123)
km = kmeans(Customers_clust, 3, nstart = 20)
km_clusters = km$cluster
km_clusters

##    [1] 2 1 2 1 3 3 3 1 1 1 1 2 3 1 2 1 3 2 1 3 3 3 3 3 1 3 1 2 1 2 1 3 2 1 3 1 1
##   [38] 2 2 1 1 1 2 1 1 3 2 2 1 2 3 2 2 1 3 2 3 3 3 2 1 1 2 2 3 2 2 3 1 1 2 2 1 3
##   [75] 1 1 1 2 2 1 1 3 2 1 1 1 1 3 1 2 2 1 1 2 2 2 1 1 3 1 2 2 2 3 2 3 1 2 3 1 1
##  [112] 3 1 1 1 2 2 1 3 3 3 2 3 2 1 1 1 1 2 3 2 3 3 3 1 1 3 1 3 3 3 1 3 2 3 1 1 2
##  [149] 1 2 1 2 1 2 1 2 1 1 1 1 1 1 2 2 1 1 2 1 1 2 1 1 1 1 3 2 1 1 2 1 3 1 2 2 2
##  [186] 3 3 2 2 1 1 1 3 1 3 3 2 3 3 2 3 1 2 3 2 1 2 3 1 3 1 3 3 2 1 3 2 1 1 2 1 1
##  [223] 2 1 2 2 1 2 3 1 2 2 2 2 1 1 3 3 1 3 3 1 1 1 3 3 1 3 1 2 3 2 1 2 1 1 1 1 3
##  [260] 2 2 2 3 1 3 3 2 1 1 2 3 2 3 1 1 2 1 1 3 1 1 2 3 1 3 1 3 1 2 1 2 3 1 1 1 2
##  [297] 3 1 1 2 1 2 1 1 3 3 2 1 1 1 1 1 1 3 1 1 2 2 1 2 2 1 3 3 1 2 1 2 1 1 2 2 2
##  [334] 2 3 1 1 2 2 3 2 3 1 1 3 3 2 1 2 3 1 3 1 3 1 1 1 1 2 1 1 1 3 3 1 1 3 2 1 3
##  [371] 2 1 2 3 3 2 3 1 1 1 1 2 3 3 3 1 1 3 1 3 1 3 3 3 2 2 1 2 2 3 1 1 1 2 1 2 2
##  [408] 1 2 2 3 3 2 1 1 2 3 1 1 1 1 1 1 1 1 3 2 1 3 2 2 1 2 3 3 2 3 3 2 3 2 1 2 1
##  [445] 2 2 1 3 3 2 1 3 1 1 3 1 3 3 3 1 1 3 1 2 2 3 3 1 1 2 3 2 3 3 3 1 2 2 2 1 1
##  [482] 1 3 1 2 2 2 1 3 1 2 3 2 3 2 1 1 2 2 1 2 3 2 1 1 2 2 2 3 3 2 3 3 1 3 1 2 3
##  [519] 1 1 3 3 2 2 1 2 1 1 1 1 1 3 1 2 3 2 2 1 2 1 2 2 2 3 3 3 1 1 1 3 1 1 3 1 3
##  [556] 1 1 1 3 1 3 1 1 1 1 2 2 3 1 1 2 2 1 2 1 1 1 1 1 1 3 2 3 1 3 1 1 1 2 3 3 1
##  [593] 1 2 3 1 1 3 3 1 2 1 2 1 2 3 1 1 2 2 2 1 2 1 2 2 3 2 2 2 3 3 2 3 3 2 3 2 1
##  [630] 3 1 3 1 3 2 3 3 1 2 1 1 1 1 1 1 3 2 3 2 2 3 1 3 2 1 2 2 2 3 3 2 2 2 2 2 3
##  [667] 2 3 1 1 1 1 1 2 2 3 2 3 2 3 2 1 3 3 1 1 3 1 2 3 2 2 1 3 1 3 3 1 2 1 1 2 2
##  [704] 3 2 1 3 3 3 2 2 2 2 1 2 3 1 1 1 2 2 1 2 1 2 2 3 2 2 2 2 3 3 1 1 1 3 2 1 2
##  [741] 3 2 2 1 3 2 2 3 1 1 1 1 2 1 2 2 1 1 1 1 1 1 3 3 3 2 2 1 3 1 3 3 3 2 1 3 1
##  [778] 3 2 2 3 1 3 3 2 1 1 2 3 2 3 1 3 3 1 2 1 3 1 2 2 3 1 2 1 1 3 3 1 1 2 3 2 1
##  [815] 3 1 3 1 1 2 2 2 3 3 1 3 3 2 1 3 3 1 3 1 2 1 3 1 3 3 3 3 1 2 1 2 2 1 1 2 2
##  [852] 3 1 2 1 1 1 1 1 2 2 3 1 3 2 1 3 2 1 2 3 2 2 3 1 2 1 2 2 3 2 2 1 1 1 2 2 3
##  [889] 1 2 2 2 2 3 2 1 2 1 1 2 3 2 2 2 2 2 1 3 1 2 1 2 3 2 3 3 3 2 2 1 3 2 3 1 3
##  [926] 1 1 3 1 1 1 3 3 1 3 2 3 1 1 3 2 2 1 1 3 2 1 1 3 2 2 2 3 1 3 1 1 1 2 2 3 2
##  [963] 2 2 1 2 1 3 2 1 1 2 1 2 3 3 2 3 1 3 3 2 2 1 1 2 1 3 1 1 2 2 3 1 1 1 1 3 1
## [1000] 1 2 1 1 1 3 2 2 2 1 2 3 1 1 1 3 2 1 3 2 1 1 1 2 3 3 2 1 2 1 1 2 3 3 2 2 3
## [1037] 3 2 1 3 1 2 3 1 2 1 2 3 1 3 2 2 1 3 1 2 3 2 1 2 3 1 2 1 2 2 1 2 1 1 2 2 2
## [1074] 1 2 2 3 3 3 1 2 1 1 2 3 2 2 1 2 1 3 1 2 1 3 2 1 3 1 3 3 3 1 1 2 3 1 1 2 2
## [1111] 3 1 2 1 3 3 1 1 1 2 1 1 3 3 1 2 2 1 2 3 1 3 2 2 2 3 3 3 2 1 3 1 1 2 2 1 1
## [1148] 2 3 1 3 1 2 1 2 3 1 3 1 1 1 1 3 1 1 2 3 1 1 1 3 1 3 2 2 3 3 3 1 2 3 2 1 1
## [1185] 1 3 2 2 2 3 1 2 1 2 1 1 1 2 1 1 2 3 1 3 1 3 1 1 1 1 3 1 2 1 1 1 1 2 2 3 1
## [1222] 1 1 1 3 2 3 2 2 2 3 3 2 1 2 1 2 2 1 1 2 3 1 1 2 2 2 3 3 1 3 1 1 2 3 2 2 1
## [1259] 3 3 1 3 1 2 2 1 1 1 1 1 3 1 3 2 1 2 1 3 2 3 3 2 2 3 3 2 3 2 3 1 3 1 1 1 2
## [1296] 3 2 3 1 3 1 1 1 2 1 3 2 2 3 3 2 1 1 3 3 3 3 1 1 1 1 2 3 2 2 2 1 3 2 2 1 2
## [1333] 2 1 3 1 3 2 3 2 3 1 3 3 3 1 1 3 3 3 3 1 3 3 3 1 2 1 1 2 3 1 1 1 1 3 1 1 3
## [1370] 3 3 3 1 3 3 3 3 2 3 3 2 2 1 2 3 3 1 2 1 1 1 2 3 1 1 1 1 1 1 3 1 1 2 1 3 1
## [1407] 3 1 1 1 1 1 1 2 2 2 2 3 2 2 3 2 2 1 3 2 3 1 2 2 3 3 1 1 1 2 3 2 1 2 1 1 1
## [1444] 3 2 1 2 1 1 3 3 3 1 1 2 2 2 3 2 3 3 3 2 3 1 2 1 3 2 2 3 1 3 3 2 2 3 3 2 3
## [1481] 2 1 2 2 1 3 3 1 1 1 2 2 1 1 1 3 2 1 2 1 3 3 3 1 1 3 1 2 3 3 1 3 2 1 3 1 1
## [1518] 1 3 1 1 2 2 2 3 1 3 1 1 2 1 3 1 2 1 3 3 2 3 3 2 1 2 1 3 3 1 2 1 3 1 2 2 1
## [1555] 2 1 3 2 3 2 1 2 1 3 1 3 1 3 2 1 3 2 2 1 1 1 2 3 3 3 2 1 2 3 1 2 1 3 3 3 1
## [1592] 1 2 3 3 1 3 2 1 1 2 3 1 1 3 3 3 2 1 1 1 3 3 1 2 1 3 2 2 1 1 2 1 1 1 3 2 3
## [1629] 2 3 3 2 1 1 1 3 1 2 2 3 2 2 2 2 1 3 1 3 1 2 1 1 1 3 1 2 3 2 2 3 2 1 1 1 2
## [1666] 1 2 1 2 3 1 1 3 1 1 3 3 2 3 2 2 1 2 1 3 2 1 3 3 1 2 2 2 3 1 3 1 3 2 1 1 1
## [1703] 2 2 3 3 2 3 1 1 3 1 2 3 2 1 2 2 2 3 3 1 3 1 1 1 1 1 2 2 1 2 2 1 2 1 2 1 1
## [1740] 1 3 2 2 1 3 1 3 1 2 1 1 2 3 3 3 3 1 3 2 1 1 2 1 3 1 3 1 3 2 2 2 1 3 1 3 3
## [1777] 1 2 2 1 3 2 2 2 1 2 2 3 1 2 1 1 2 2 1 3 2 2 1 1 1 3 3 3 1 2 3 3 3 1 2 1 2
## [1814] 3 2 1 1 3 1 2 1 2 2 2 3 1 1 2 3 2 1 2 2 2 3 1 1 3 2 1 2 1 3 1 1 3 2 2 2 2
## [1851] 1 1 2 1 3 1 2 3 3 2 2 1 3 2 3 1 2 3 1 1 2 1 1 2 2 1 1 1 2 2 2 2 3 3 1 1 1
## [1888] 1 1 2 2 2 2 1 3 2 2 3 1 3 1 1 2 1 3 3 1 1 2 1 1 2 1 2 2 2 1 1 2 2 2 3 2 1
## [1925] 3 2 1 2 3 1 3 1 2 2 2 2 2 3 1 3 3 3 3 2 1 1 1 1 1 2 1 3 1 3 2 1 2 2 1 2 2
## [1962] 1 2 2 3 3 2 1 1 1 1 1 1 2 3 3 3 2 1 2 3 2 1 1 1 1 1 1 1 3 3 2 1 1 1 3 1 3
## [1999] 3 1 2 3 2 3 2 1 2 2 3 1 1 3 3 1 2 2 2 1 3 2 1 1 1 2 2 3 1 2 1 3 2 1 3 1 1
## [2036] 3 2 1 2 2 2 1 1 1 3 2 2 2 2 1 1 3 1 2 2 3 1 2 3 2 2 1 2 1 2 1 1 1 3 3 2 3
## [2073] 2 1 1 3 2 1 2 2 1 1 1 1 3 1 2 3 2 3 1 1 2 1 2 3 3 1 1 3 1 1 1 1 3 2 1 1 3
## [2110] 1 1 2 1 3 1 1 3 1 3 3 2 3 3 2 1 1 2 2 2 1 3 2 2 2 3 2 2 3 3 3 1 3 1 1 1 2
## [2147] 2 2 2 1 2 1 1 2 2 1 1 3 3 1 1 3 3 2 1 3 3 1 1 1 3 2 1 2 1 1 1 2 1 3 3 2 1
## [2184] 1 3 3 3 3 1 1 3 1 1 2 3 2 2 3

Compare K-Means and Hierarchical Clustering

table(km_clusters, hc_complete)

##            hc_complete
## km_clusters   1   2   3
##           1  24 832  20
##           2 700   0   0
##           3 272  62 288

hc_clusters = hc_complete

Cluster visualisation

Append the result to our original dataset indicating which cluster each customer belongs to.

Customers_cleaned["kmcluster"] <- km_clusters
Customers_cleaned["hcluster"] <- hc_clusters
Customers_cleaned

## # A tibble: 2,198 × 37
##    Education Income Recency MntWines MntFruits MntMeatProducts MntFishProducts
##        <dbl>  <dbl>   <dbl>    <dbl>     <dbl>           <dbl>           <dbl>
##  1         2  58138      58      635        88             546             172
##  2         2  46344      38       11         1               6               2
##  3         2  71613      26      426        49             127             111
##  4         2  26646      26       11         4              20              10
##  5         4  58293      94      173        43             118              46
##  6         3  62513      16      520        42              98               0
##  7         2  55635      34      235        65             164              50
##  8         4  33454      32       76        10              56               3
##  9         4  30351      19       14         0              24               3
## 10         4   5648      68       28         0               6               1
## # ℹ 2,188 more rows
## # ℹ 30 more variables: MntSweetProducts <dbl>, MntGoldProds <dbl>,
## #   NumDealsPurchases <dbl>, NumWebPurchases <dbl>, NumCatalogPurchases <dbl>,
## #   NumStorePurchases <dbl>, NumWebVisitsMonth <dbl>, AcceptedCmp3 <dbl>,
## #   AcceptedCmp4 <dbl>, AcceptedCmp5 <dbl>, AcceptedCmp1 <dbl>,
## #   AcceptedCmp2 <dbl>, Complain <dbl>, Response <dbl>, Age <dbl>,
## #   AgeCategory <dbl>, NumChildren <dbl>, Spending <dbl>, log_Wines <dbl>, …

PC1 <- Customers_clust[,1]
PC2 <- Customers_clust[,2]
PC3 <- Customers_clust[,3]

# Append the cluster result to our dataset
Customers_clust["hcluster"] <- hc_clusters
cluster3d <- plot_ly(Customers_clust, x = ~Customers_clust$PC1, y = ~Customers_clust$PC2, z = ~Customers_clust$PC3, color = ~as.factor(Customers_cleaned$hcluster), colors = c('#636EFA','#EF553B','#00CC96') ) %>%
  add_markers(size = 12)


cluster3d <- cluster3d %>%
  layout(
    scene = list(
      bgcolor = "#e5ecf6",
      xaxis = list(title = "PC1"),
      yaxis = list(title = "PC2"),
      zaxis = list(title = "PC3")
    )
  )

cluster3d

Results Interpretation

Compare clustering results between K-Means and Hierarchical methods

Hierarchical Clustering Summary statistics

hcresult <- Customers_cleaned %>%
  group_by(hcluster) %>%
  summarise_all(mean) %>%
  t() %>%
  round(2)
hcresult

##                         [,1]     [,2]     [,3]
## hcluster                1.00     2.00     3.00
## Education               2.57     2.35     2.57
## Income              68695.69 32949.33 49606.84
## Recency                49.24    48.38    50.06
## MntWines              538.45    34.30   337.78
## MntFruits              49.72     4.06    15.06
## MntMeatProducts       314.15    18.36   109.38
## MntFishProducts        71.37     5.70    21.64
## MntSweetProducts       50.76     4.19    15.63
## MntGoldProds           66.53    13.13    60.17
## NumDealsPurchases       1.79     1.89     5.32
## NumWebPurchases         5.37     1.91     6.22
## NumCatalogPurchases     4.72     0.44     2.24
## NumStorePurchases       8.22     3.08     5.99
## NumWebVisitsMonth       3.71     6.53     7.20
## AcceptedCmp3            0.07     0.07     0.09
## AcceptedCmp4            0.12     0.01     0.09
## AcceptedCmp5            0.16     0.00     0.00
## AcceptedCmp1            0.13     0.00     0.02
## AcceptedCmp2            0.03     0.00     0.00
## Complain                0.01     0.01     0.00
## Response                0.20     0.08     0.19
## Age                    54.10    49.10    54.14
## AgeCategory             2.28     2.01     2.32
## NumChildren             0.52     1.23     1.54
## Spending             1090.99    79.73   559.65
## log_Wines               6.05     2.87     5.53
## log_Fruits              3.34     1.16     1.86
## log_MeatProducts        5.42     2.62     4.36
## log_FishProducts        3.69     1.40     2.15
## log_SweetProducts       3.34     1.18     1.79
## log_GoldProds           3.80     2.19     3.67
## log_Spending            6.85     4.08     6.13
## Relationship            1.64     1.65     1.65
## YearsJoined             7.95     7.83     8.44
## TotalAcceptedCmp        0.71     0.17     0.40
## kmcluster               2.25     1.14     2.87

K-Means Summary Statistics

kmresult <- Customers_cleaned %>%
  group_by(kmcluster) %>%
  summarise_all(mean) %>%
  t() %>%
  round(2)
kmresult

##                         [,1]     [,2]     [,3]
## kmcluster               1.00     2.00     3.00
## Education               2.37     2.47     2.65
## Income              32699.80 73300.87 53377.64
## Recency                48.41    49.14    49.70
## MntWines               27.60   584.81   381.75
## MntFruits               3.85    62.85    16.76
## MntMeatProducts        16.01   397.85   113.31
## MntFishProducts         5.45    90.97    23.13
## MntSweetProducts        4.00    64.17    17.19
## MntGoldProds           12.21    74.90    53.71
## NumDealsPurchases       1.85     1.40     4.05
## NumWebPurchases         1.79     5.26     5.98
## NumCatalogPurchases     0.40     5.53     2.52
## NumStorePurchases       2.99     8.52     6.76
## NumWebVisitsMonth       6.50     3.05     6.31
## AcceptedCmp3            0.07     0.08     0.08
## AcceptedCmp4            0.01     0.11     0.12
## AcceptedCmp5            0.00     0.21     0.02
## AcceptedCmp1            0.00     0.17     0.03
## AcceptedCmp2            0.00     0.03     0.01
## Complain                0.01     0.01     0.01
## Response                0.08     0.24     0.15
## Age                    48.98    53.08    55.29
## AgeCategory             2.00     2.22     2.37
## NumChildren             1.24     0.32     1.25
## Spending               69.12  1275.55   605.86
## log_Wines               2.79     6.20     5.64
## log_Fruits              1.11     3.76     2.14
## log_MeatProducts        2.56     5.79     4.48
## log_FishProducts        1.36     4.17     2.37
## log_SweetProducts       1.14     3.76     2.09
## log_GoldProds           2.15     3.95     3.57
## log_Spending            4.02     7.08     6.24
## Relationship            1.65     1.63     1.65
## YearsJoined             7.83     7.95     8.19
## TotalAcceptedCmp        0.16     0.83     0.41
## hcluster                2.00     1.00     2.03

From this, we can safely use the clustering result from Hierarchical Clustering.

Number of customers in each cluster

barplot(table(hc_complete), main = "Number of customers in each cluster", xlab = "Clusters", ylab = "Counts")

Education level

Customers_cleaned$Education <- as.factor(Customers_cleaned$Education) # change in categorical variable 
ggplot(Customers_cleaned, aes(x = hcluster, fill = Education)) +
  geom_bar(position = "fill") +
  labs(title = "Bar plot of education by cluster", x = "hcluster", y = "proportion") 

Income level

mean_result <- aggregate(data = Customers_cleaned, Income ~ hcluster, mean)
sd_result <- aggregate(data = Customers_cleaned, Income ~ hcluster, sd)
count_result <- table(Customers_cleaned$hcluster)

# Combine mean, sd, and count into a single table
combined_table <- merge(merge(mean_result, sd_result, by = "hcluster"), as.data.frame(count_result), by.x = "hcluster", by.y = "Var1", all.x = TRUE)

# Rename the columns 
colnames(combined_table) <- c("hcluster", "estimated_income", "sd_Income", "sample_count")

#calculate the standard error
combined_table$standard_error <- combined_table$sd_Income/ sqrt(combined_table$sample_count)

#calculate the t_score using 95% confidence interval 
alpha = 0.05
degrees_of_freedom = combined_table$sample_count - 1
combined_table$t_score = qt(p=alpha/2, df=degrees_of_freedom,lower.tail=F)

#calculate the margin of error
combined_table$margin_error <- combined_table$t_score * combined_table$standard_error

#show the table result
combined_table

##   hcluster estimated_income sd_Income sample_count standard_error  t_score
## 1        1         68695.69  12793.98          996       405.3928 1.962351
## 2        2         32949.33  11696.60          894       391.1928 1.962624
## 3        3         49606.84  11566.32          308       659.0524 1.967721
##   margin_error
## 1     795.5229
## 2     767.7643
## 3    1296.8314

ggplot(combined_table, aes(x = estimated_income, y = reorder(hcluster, estimated_income))) + 
  geom_errorbarh(aes(xmin = estimated_income - margin_error, xmax = estimated_income + margin_error)) + 
  geom_point(size = 3, color = "darkgreen") + 
  theme_minimal(base_size = 12.5) + 
  labs(title = "Mean customer household income", 
       subtitle = "For Each Hierarchial Cluster", 
       x = "Income Estimate", 
       y = "Cluster group")

Num of Children

numchildren <- Customers_cleaned %>%
  group_by(hcluster) %>%
  summarise(across(c("NumChildren"), mean))

numchildren_ct <- numchildren %>% gather(key = numchildren, value = Value, NumChildren)

ggplot(numchildren_ct, aes(x = factor(hcluster), y = Value, fill = numchildren)) +
  geom_bar(stat = "identity", position = "dodge") +
  labs(x = "Cluster Groups", y = "Total", title = "Average no. of Children") +
  theme_minimal()

Spending amount

# Create a scatterplot
spending_plot <- ggplot(Customers_cleaned, aes(x = Spending, y = Income, color = as.factor(hcluster))) +
  geom_point() +
  labs(title = "Income and Spending for each Cluster") +
  theme_minimal()

# Show the plot
print(spending_plot)

Spending patterns

goods <- Customers_cleaned %>%
  select(MntWines, MntFruits, MntMeatProducts, MntFishProducts, MntSweetProducts, MntGoldProds, Spending, hcluster)

sum_spending_by_commodity <- goods %>%
  group_by(hcluster) %>%
  summarise(across(c("MntWines", "MntFruits", "MntMeatProducts", "MntFishProducts", "MntSweetProducts", "MntGoldProds"), sum))

spending_by_cluster <- goods %>%
  group_by(hcluster) %>%
  summarise(across(c("Spending"), sum))

proportion <- sum_spending_by_commodity %>%
  mutate(across(-1, ~./spending_by_cluster$Spending))

library(gt)
library(scales)

## 
## Attaching package: 'scales'

## The following object is masked from 'package:readr':
## 
##     col_factor

spend_table <- proportion %>%
  gt() %>%
  data_color(
    columns = c("MntWines", "MntFruits", "MntMeatProducts", "MntFishProducts", "MntSweetProducts", "MntGoldProds"),
    colors = scales::col_numeric(
      palette = "YlGn",
      domain = NULL
    ) 
  )

## Warning: Since gt v0.9.0, the `colors` argument has been deprecated.
## • Please use the `fn` argument instead.
## This warning is displayed once every 8 hours.

spend_table

hcluster	MntWines	MntFruits	MntMeatProducts	MntFishProducts	MntSweetProducts	MntGoldProds
1	0.4935451	0.04557602	0.2879533	0.06541545	0.04652943	0.06098062
2	0.4301647	0.05089644	0.2302405	0.07149070	0.05249572	0.16471199
3	0.6035493	0.02690692	0.1954494	0.03866637	0.02792217	0.10750586

Campaigns performance

campaigns_plot <- ggplot(Customers_cleaned, aes(x = factor(TotalAcceptedCmp), fill = as.factor(hcluster))) +
  geom_bar(position = "dodge") +
  labs(title = "Count Of Promotion Accepted",
       x = "Number Of Total Accepted Promotions") +
  theme_minimal()

campaigns_plot

Deals purchased

deals_plot <- ggplot(Customers_cleaned, aes(x = NumDealsPurchases, fill = as.factor(hcluster))) +
  geom_bar(position = "dodge") +
  labs(title = "Count of Deals Puchased",
       x = "Number Of Deals Purhcased") +
  theme_minimal()

deals_plot

Sales channels

sum_spending_by_commodity <- Customers_cleaned %>%
  group_by(hcluster) %>%
  summarise(across(c("NumWebPurchases", "NumCatalogPurchases", "NumStorePurchases"), mean))

purchases_ct <- sum_spending_by_commodity %>% gather(key = Purchases, value = Value, NumWebPurchases:NumStorePurchases)

ggplot(purchases_ct, aes(x = factor(hcluster), y = Value, fill = Purchases)) +
  geom_bar(stat = "identity", position = "dodge") +
  labs(x = "Cluster Groups", y = "Average Counts", title = "Average Counts of Purchases by Sales Channels") +
  scale_fill_manual(values = c("NumWebPurchases" = "blue", "NumCatalogPurchases" = "green", "NumStorePurchases" = "red")) +
  theme_minimal()

Supervised classification problem

Classifying the clusters

To maintain the consistency of our analysis, we should use the same variables that we used in the clustering analysis.

Customers_PCA["hcluster"] <- hc_clusters

Customers_PCA <- Customers_PCA %>% 
  mutate( 
    InCluster1 = as.factor(if_else(hcluster == 1, 1, 0)), 
    InCluster2 = as.factor(if_else(hcluster == 2, 1, 0)), 
    InCluster3 = as.factor(if_else(hcluster == 3, 1, 0))) 

Tree for classifying cluster 1

# Fit a tree for cluster 1
tree1.Customers = tree(formula = InCluster1~.-InCluster2-InCluster3-hcluster, data=Customers_PCA) 
plot(tree1.Customers) 
text(tree1.Customers, pretty=1, cex = 0.7) 

cv1.Customers = cv.tree(tree1.Customers, FUN=prune.misclass)
cv1.Customers$size

## [1] 10  9  8  6  3  2  1

cv1.Customers$dev

## [1] 177 177 190 190 231 265 996

# Prune the tree for cluster 1
prune1.Customers = prune.misclass(tree1.Customers, best=6)
plot(prune1.Customers)
text(prune1.Customers, pretty=1, cex = 0.7) 

Tree for classifying cluster 2

# Fit the tree for cluster 2
tree2.Customers = tree(formula = InCluster2~.-InCluster1-InCluster3-hcluster, data=Customers_PCA) 
plot(tree2.Customers) 
text(tree2.Customers, pretty=1, cex = 0.7) 

cv2.Customers = cv.tree(tree2.Customers, FUN=prune.misclass)
cv2.Customers$size

## [1] 5 2 1

cv2.Customers$dev

## [1] 118 118 894

# Prune the tree for cluster 2
prune2.Customers = prune.misclass(tree2.Customers, best=2)
plot(prune2.Customers)
text(prune2.Customers, pretty=1, cex = 0.7) 

Tree for classifying cluster 3

# Fit the tree for cluster 3
tree3.Customers = tree(formula = InCluster3~.-InCluster1-InCluster2-hcluster, data=Customers_PCA) 
plot(tree3.Customers) 
text(tree3.Customers, pretty=1, cex = 0.7) 

cv3.Customers = cv.tree(tree3.Customers, FUN=prune.misclass)
cv3.Customers$size

## [1] 14 10  8  5  4  3  2  1

cv3.Customers$dev

## [1] 126 126 130 137 141 195 229 308

# Prune the tree for cluster 3
prune3.Customers = prune.misclass(tree3.Customers, best=5)
plot(prune3.Customers)
text(prune3.Customers, pretty=1, cex = 0.7) 

Predict the response variable

sum(Customers_cleaned$Response == 1)

## [1] 329

sum(Customers_cleaned$Response == 0)

## [1] 1869

glm.fits = glm(Response ~ Education + Income + NumChildren + Recency + NumDealsPurchases + NumWebPurchases + NumStorePurchases + YearsJoined + Relationship + Age + Spending + TotalAcceptedCmp, data = Customers_cleaned, family = binomial)
summary(glm.fits)

## 
## Call:
## glm(formula = Response ~ Education + Income + NumChildren + Recency + 
##     NumDealsPurchases + NumWebPurchases + NumStorePurchases + 
##     YearsJoined + Relationship + Age + Spending + TotalAcceptedCmp, 
##     family = binomial, data = Customers_cleaned)
## 
## Coefficients:
##                     Estimate Std. Error z value Pr(>|z|)    
## (Intercept)       -1.253e+01  1.708e+00  -7.333 2.25e-13 ***
## Education2         7.579e-01  4.370e-01   1.734 0.082875 .  
## Education3         1.093e+00  4.895e-01   2.232 0.025598 *  
## Education4         1.658e+00  4.721e-01   3.512 0.000444 ***
## Income             6.088e-07  1.159e-05   0.053 0.958093    
## NumChildren       -6.878e-01  2.262e-01  -3.040 0.002363 ** 
## Recency           -3.798e-02  4.534e-03  -8.376  < 2e-16 ***
## NumDealsPurchases  1.917e-01  6.604e-02   2.902 0.003704 ** 
## NumWebPurchases    6.584e-02  4.733e-02   1.391 0.164188    
## NumStorePurchases -2.446e-01  5.250e-02  -4.659 3.18e-06 ***
## YearsJoined        1.525e+00  1.849e-01   8.249  < 2e-16 ***
## Relationship      -1.534e+00  2.339e-01  -6.557 5.49e-11 ***
## Age                9.480e-04  9.792e-03   0.097 0.922872    
## Spending          -4.803e-04  4.020e-04  -1.195 0.232139    
## TotalAcceptedCmp   3.450e+00  2.080e-01  16.584  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1855.80  on 2197  degrees of freedom
## Residual deviance:  572.21  on 2183  degrees of freedom
## AIC: 602.21
## 
## Number of Fisher Scoring iterations: 7

We now remove the insignificant variables one by one

glm.fits1 = glm(Response ~ Education + NumChildren + Recency + NumDealsPurchases + NumWebPurchases + NumStorePurchases + YearsJoined + Relationship + Age + Spending + TotalAcceptedCmp, data = Customers_cleaned, family = binomial)
summary(glm.fits1)

## 
## Call:
## glm(formula = Response ~ Education + NumChildren + Recency + 
##     NumDealsPurchases + NumWebPurchases + NumStorePurchases + 
##     YearsJoined + Relationship + Age + Spending + TotalAcceptedCmp, 
##     family = binomial, data = Customers_cleaned)
## 
## Coefficients:
##                     Estimate Std. Error z value Pr(>|z|)    
## (Intercept)       -1.250e+01  1.624e+00  -7.696 1.41e-14 ***
## Education2         7.613e-01  4.322e-01   1.762 0.078136 .  
## Education3         1.095e+00  4.866e-01   2.251 0.024392 *  
## Education4         1.661e+00  4.688e-01   3.543 0.000395 ***
## NumChildren       -6.850e-01  2.200e-01  -3.114 0.001845 ** 
## Recency           -3.798e-02  4.534e-03  -8.377  < 2e-16 ***
## NumDealsPurchases  1.906e-01  6.307e-02   3.023 0.002506 ** 
## NumWebPurchases    6.613e-02  4.691e-02   1.410 0.158625    
## NumStorePurchases -2.439e-01  5.071e-02  -4.809 1.52e-06 ***
## YearsJoined        1.523e+00  1.802e-01   8.448  < 2e-16 ***
## Relationship      -1.534e+00  2.338e-01  -6.560 5.37e-11 ***
## Age                1.008e-03  9.724e-03   0.104 0.917423    
## Spending          -4.655e-04  2.872e-04  -1.621 0.104980    
## TotalAcceptedCmp   3.450e+00  2.080e-01  16.591  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1855.80  on 2197  degrees of freedom
## Residual deviance:  572.22  on 2184  degrees of freedom
## AIC: 600.22
## 
## Number of Fisher Scoring iterations: 7

glm.fits2 = glm(Response ~ Education + NumChildren + Recency + NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + Age + Spending + TotalAcceptedCmp, data = Customers_cleaned, family = binomial)
summary(glm.fits2)

## 
## Call:
## glm(formula = Response ~ Education + NumChildren + Recency + 
##     NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + 
##     Age + Spending + TotalAcceptedCmp, family = binomial, data = Customers_cleaned)
## 
## Coefficients:
##                     Estimate Std. Error z value Pr(>|z|)    
## (Intercept)       -1.250e+01  1.610e+00  -7.765 8.15e-15 ***
## Education2         7.443e-01  4.294e-01   1.733 0.083054 .  
## Education3         1.055e+00  4.832e-01   2.183 0.029008 *  
## Education4         1.670e+00  4.658e-01   3.586 0.000336 ***
## NumChildren       -6.846e-01  2.184e-01  -3.135 0.001719 ** 
## Recency           -3.790e-02  4.537e-03  -8.353  < 2e-16 ***
## NumDealsPurchases  2.118e-01  6.035e-02   3.510 0.000449 ***
## NumStorePurchases -2.339e-01  5.009e-02  -4.671 3.00e-06 ***
## YearsJoined        1.527e+00  1.787e-01   8.547  < 2e-16 ***
## Relationship      -1.528e+00  2.330e-01  -6.560 5.39e-11 ***
## Age                2.586e-03  9.658e-03   0.268 0.788881    
## Spending          -3.580e-04  2.739e-04  -1.307 0.191235    
## TotalAcceptedCmp   3.445e+00  2.069e-01  16.652  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1855.80  on 2197  degrees of freedom
## Residual deviance:  574.11  on 2185  degrees of freedom
## AIC: 600.11
## 
## Number of Fisher Scoring iterations: 7

glm.fits3 = glm(Response ~ Education + NumChildren + Recency + NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + Spending + TotalAcceptedCmp, data = Customers_cleaned, family = binomial)
summary(glm.fits3)

## 
## Call:
## glm(formula = Response ~ Education + NumChildren + Recency + 
##     NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + 
##     Spending + TotalAcceptedCmp, family = binomial, data = Customers_cleaned)
## 
## Coefficients:
##                     Estimate Std. Error z value Pr(>|z|)    
## (Intercept)       -1.238e+01  1.545e+00  -8.015 1.10e-15 ***
## Education2         7.572e-01  4.265e-01   1.775 0.075831 .  
## Education3         1.072e+00  4.792e-01   2.236 0.025329 *  
## Education4         1.685e+00  4.623e-01   3.645 0.000267 ***
## NumChildren       -6.749e-01  2.156e-01  -3.131 0.001744 ** 
## Recency           -3.794e-02  4.538e-03  -8.361  < 2e-16 ***
## NumDealsPurchases  2.112e-01  6.038e-02   3.498 0.000468 ***
## NumStorePurchases -2.331e-01  4.997e-02  -4.664 3.11e-06 ***
## YearsJoined        1.526e+00  1.787e-01   8.543  < 2e-16 ***
## Relationship      -1.533e+00  2.324e-01  -6.598 4.15e-11 ***
## Spending          -3.488e-04  2.717e-04  -1.284 0.199204    
## TotalAcceptedCmp   3.444e+00  2.069e-01  16.645  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1855.80  on 2197  degrees of freedom
## Residual deviance:  574.18  on 2186  degrees of freedom
## AIC: 598.18
## 
## Number of Fisher Scoring iterations: 7

glm.fits4 = glm(Response ~ Education + NumChildren + Recency + NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + TotalAcceptedCmp, data = Customers_cleaned, family = binomial)
summary(glm.fits4)

## 
## Call:
## glm(formula = Response ~ Education + NumChildren + Recency + 
##     NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + 
##     TotalAcceptedCmp, family = binomial, data = Customers_cleaned)
## 
## Coefficients:
##                     Estimate Std. Error z value Pr(>|z|)    
## (Intercept)       -12.252265   1.537240  -7.970 1.58e-15 ***
## Education2          0.694876   0.422692   1.644 0.100191    
## Education3          1.010858   0.475816   2.124 0.033630 *  
## Education4          1.630891   0.458252   3.559 0.000372 ***
## NumChildren        -0.567129   0.196436  -2.887 0.003888 ** 
## Recency            -0.038237   0.004516  -8.467  < 2e-16 ***
## NumDealsPurchases   0.208783   0.059274   3.522 0.000428 ***
## NumStorePurchases  -0.266813   0.042675  -6.252 4.05e-10 ***
## YearsJoined         1.509965   0.177595   8.502  < 2e-16 ***
## Relationship       -1.538676   0.231947  -6.634 3.27e-11 ***
## TotalAcceptedCmp    3.369045   0.195940  17.194  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1855.80  on 2197  degrees of freedom
## Residual deviance:  575.86  on 2187  degrees of freedom
## AIC: 597.86
## 
## Number of Fisher Scoring iterations: 7

Predicting the ‘Reponse’ variable using all the data

glm.probs = predict(glm.fits4, type = "response")
glm.pred = rep(0, 2198)
glm.pred[glm.probs > 0.5] = 1
table(glm.pred, as.factor(Customers_cleaned$Response))

##         
## glm.pred    0    1
##        0 1821   77
##        1   48  252

mean(glm.pred == Customers_cleaned$Response)

## [1] 0.9431301

Our logistic regression model correctly predicts the response variable 94.3% of the time.

We now split into training and testing data

set.seed(123)

# Subset a dataframe with only response = 1 
Customers_cleaned_Response1 <- subset(Customers_cleaned, Response == 1)
Customers_cleaned_Response0 <- subset(Customers_cleaned, Response == 0)
dim(Customers_cleaned_Response1) #329 

## [1] 329  37

# We subset a dataframe with only response = 0 and then random sample into equal size with response = 1
Customers_cleaned_Response0_ind <- sample(1:nrow(Customers_cleaned_Response0), nrow(Customers_cleaned_Response1)) 
Customers_cleaned_Response0 <- Customers_cleaned_Response0[Customers_cleaned_Response0_ind, ]

# Merge 
train <- rbind(Customers_cleaned_Response1, Customers_cleaned_Response0)
train_ind <- sample(1:nrow(train), nrow(train)*0.75) 
train <- Customers_cleaned[train_ind, ]

# Extract the remaining observations as testing data
test <- Customers_cleaned[-train_ind, ]

# Re-fit the logistic regression model 
glm.fits5 = glm(Response ~ Education + NumChildren + Recency + NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + TotalAcceptedCmp, data = train, family = binomial)

# Result
summary(glm.fits5)

## 
## Call:
## glm(formula = Response ~ Education + NumChildren + Recency + 
##     NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + 
##     TotalAcceptedCmp, family = binomial, data = train)
## 
## Coefficients:
##                    Estimate Std. Error z value Pr(>|z|)    
## (Intercept)       -14.28008    4.14399  -3.446 0.000569 ***
## Education2          1.48484    1.01416   1.464 0.143163    
## Education3          2.37658    1.21590   1.955 0.050631 .  
## Education4          2.10854    1.21319   1.738 0.082208 .  
## NumChildren        -1.32332    0.61756  -2.143 0.032129 *  
## Recency            -0.04154    0.01341  -3.099 0.001942 ** 
## NumDealsPurchases   0.28165    0.15016   1.876 0.060694 .  
## NumStorePurchases  -0.40243    0.12095  -3.327 0.000877 ***
## YearsJoined         1.76272    0.47385   3.720 0.000199 ***
## Relationship       -1.88713    0.67219  -2.807 0.004994 ** 
## TotalAcceptedCmp    4.42919    0.62431   7.095  1.3e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 384.422  on 492  degrees of freedom
## Residual deviance:  87.701  on 482  degrees of freedom
## AIC: 109.7
## 
## Number of Fisher Scoring iterations: 8

# Prediction
glm.probs2 = predict(glm.fits5, newdata = test, type = "response")
glm.pred2 <- ifelse(glm.probs2 > 0.5, 1, 0)
table(glm.pred2, as.factor(test$Response))

##          
## glm.pred2    0    1
##         0 1397   64
##         1   44  200

# Accuracy 
mean(glm.pred2 == test$Response)

## [1] 0.9366569

Association Rule

Load relevant libraries

library(arules)

## Loading required package: Matrix

## 
## Attaching package: 'Matrix'

## The following objects are masked from 'package:tidyr':
## 
##     expand, pack, unpack

## 
## Attaching package: 'arules'

## The following object is masked from 'package:dplyr':
## 
##     recode

## The following objects are masked from 'package:base':
## 
##     abbreviate, write

library(effects)

## Loading required package: carData

## lattice theme set by effectsTheme()
## See ?effectsTheme for details.

library(arulesViz)

1. Web sales channel

# Extract the relevant features
CustomersAssoc1 <- Customers_cleaned[c("AgeCategory", "NumChildren", "Education", "Relationship", "Recency", "YearsJoined", "NumWebPurchases")]

# Convert into categorical variables using quantiles
CustomersAssoc1$NumWebPurchases <- cut(
  CustomersAssoc1$NumWebPurchases,
  breaks = quantile(CustomersAssoc1$NumWebPurchases, c(0, 0.33, 0.66, 1)),
  labels = c("Low", "Medium", "High"),
  include.lowest = TRUE
)

CustomersAssoc1$Recency <- cut(
  CustomersAssoc1$Recency,
  breaks = quantile(CustomersAssoc1$Recency, c(0, 0.5, 1)),
  labels = c("Recent", "Distant"),
  include.lowest = TRUE
)

CustomersAssoc1$YearsJoined <- cut(
  CustomersAssoc1$YearsJoined,
  breaks = quantile(CustomersAssoc1$YearsJoined, c(0, 0.5, 1)),
  labels = c("<8 years", ">8 years"),
  include.lowest = TRUE
)

CustomersAssoc1$NumChildren <- ifelse(CustomersAssoc1$NumChildren == 0,"NoChild", "YesChild")

CustomersAssoc1$AgeCategory <- ifelse(CustomersAssoc1$AgeCategory == 1,"Millenials", 
                                  ifelse(CustomersAssoc1$AgeCategory == 2, "GenX", "BabyBoomer"))

CustomersAssoc1$Relationship <- ifelse(CustomersAssoc1$Relationship == 1,"Not-Partnered", "Partnered")

CustomersAssoc1$Education <- ifelse(CustomersAssoc1$Education == 1,"Bachelors",
                                    ifelse(CustomersAssoc1$Education == 2, "Graduates",
                                           ifelse(CustomersAssoc1$Education ==3, "Masters", "PhDs")))

# Association rules for web channel
rules_web <- apriori(CustomersAssoc1, parameter = list(support = 0.05, confidence = 0.3), appearance = list(rhs = "NumWebPurchases=High"))

## Warning: Column(s) 1, 2, 3, 4 not logical or factor. Applying default
## discretization (see '? discretizeDF').

## Apriori
## 
## Parameter specification:
##  confidence minval smax arem  aval originalSupport maxtime support minlen
##         0.3    0.1    1 none FALSE            TRUE       5    0.05      1
##  maxlen target  ext
##      10  rules TRUE
## 
## Algorithmic control:
##  filter tree heap memopt load sort verbose
##     0.1 TRUE TRUE  FALSE TRUE    2    TRUE
## 
## Absolute minimum support count: 109 
## 
## set item appearances ...[1 item(s)] done [0.00s].
## set transactions ...[18 item(s), 2198 transaction(s)] done [0.00s].
## sorting and recoding items ... [18 item(s)] done [0.00s].
## creating transaction tree ... done [0.00s].
## checking subsets of size 1 2 3 4 5 6 done [0.00s].
## writing ... [17 rule(s)] done [0.00s].
## creating S4 object  ... done [0.00s].

inspect(rules_web)

##      lhs                          rhs                       support confidence  coverage     lift count
## [1]  {Education=PhDs}          => {NumWebPurchases=High} 0.07051865  0.3283898 0.2147407 1.171755   155
## [2]  {YearsJoined=>8 years}    => {NumWebPurchases=High} 0.08007279  0.3606557 0.2220200 1.286885   176
## [3]  {AgeCategory=BabyBoomer}  => {NumWebPurchases=High} 0.11692448  0.3346354 0.3494086 1.194040   257
## [4]  {NumChildren=YesChild,                                                                            
##       Education=PhDs}          => {NumWebPurchases=High} 0.05186533  0.3294798 0.1574158 1.175644   114
## [5]  {Education=PhDs,                                                                                  
##       YearsJoined=<8 years}    => {NumWebPurchases=High} 0.05050045  0.3016304 0.1674249 1.076272   111
## [6]  {Relationship=Partnered,                                                                          
##       YearsJoined=>8 years}    => {NumWebPurchases=High} 0.05277525  0.3741935 0.1410373 1.335191   116
## [7]  {NumChildren=YesChild,                                                                            
##       YearsJoined=>8 years}    => {NumWebPurchases=High} 0.05914468  0.3757225 0.1574158 1.340646   130
## [8]  {AgeCategory=BabyBoomer,                                                                          
##       Recency=Distant}         => {NumWebPurchases=High} 0.05914468  0.3194103 0.1851683 1.139714   130
## [9]  {AgeCategory=BabyBoomer,                                                                          
##       Recency=Recent}          => {NumWebPurchases=High} 0.05777980  0.3518006 0.1642402 1.255288   127
## [10] {AgeCategory=BabyBoomer,                                                                          
##       Education=Graduates}     => {NumWebPurchases=High} 0.05095541  0.3043478 0.1674249 1.085968   112
## [11] {AgeCategory=BabyBoomer,                                                                          
##       Relationship=Partnered}  => {NumWebPurchases=High} 0.07142857  0.3230453 0.2211101 1.152684   157
## [12] {AgeCategory=BabyBoomer,                                                                          
##       NumChildren=YesChild}    => {NumWebPurchases=High} 0.08553230  0.3686275 0.2320291 1.315330   188
## [13] {AgeCategory=BabyBoomer,                                                                          
##       YearsJoined=<8 years}    => {NumWebPurchases=High} 0.08644222  0.3130148 0.2761601 1.116894   190
## [14] {AgeCategory=BabyBoomer,                                                                          
##       NumChildren=YesChild,                                                                            
##       Relationship=Partnered}  => {NumWebPurchases=High} 0.05641492  0.3746224 0.1505914 1.336721   124
## [15] {AgeCategory=BabyBoomer,                                                                          
##       Relationship=Partnered,                                                                          
##       YearsJoined=<8 years}    => {NumWebPurchases=High} 0.05368517  0.3072917 0.1747043 1.096473   118
## [16] {AgeCategory=BabyBoomer,                                                                          
##       NumChildren=YesChild,                                                                            
##       YearsJoined=<8 years}    => {NumWebPurchases=High} 0.06278435  0.3424318 0.1833485 1.221859   138
## [17] {NumChildren=YesChild,                                                                            
##       Relationship=Partnered,                                                                          
##       Recency=Distant}         => {NumWebPurchases=High} 0.07370337  0.3079848 0.2393085 1.098946   162

2. Store sales channel

# Subset the relevant variables for the second association rule
CustomersAssoc2 <- subset(CustomersAssoc1, select = -NumWebPurchases)
CustomersAssoc2$NumStorePurchases <- Customers_cleaned$NumStorePurchases

CustomersAssoc2$NumStorePurchases <- cut(
  CustomersAssoc2$NumStorePurchases,
  breaks = quantile(CustomersAssoc2$NumStorePurchases, c(0, 0.33, 0.66, 1)),
  labels = c("Low", "Medium", "High"),
  include.lowest = TRUE
)

rules_store <- apriori(CustomersAssoc2, parameter = list(support = 0.05, confidence = 0.4), appearance = list(rhs = "NumStorePurchases=High"))

## Warning: Column(s) 1, 2, 3, 4 not logical or factor. Applying default
## discretization (see '? discretizeDF').

## Apriori
## 
## Parameter specification:
##  confidence minval smax arem  aval originalSupport maxtime support minlen
##         0.4    0.1    1 none FALSE            TRUE       5    0.05      1
##  maxlen target  ext
##      10  rules TRUE
## 
## Algorithmic control:
##  filter tree heap memopt load sort verbose
##     0.1 TRUE TRUE  FALSE TRUE    2    TRUE
## 
## Absolute minimum support count: 109 
## 
## set item appearances ...[1 item(s)] done [0.00s].
## set transactions ...[18 item(s), 2198 transaction(s)] done [0.00s].
## sorting and recoding items ... [18 item(s)] done [0.00s].
## creating transaction tree ... done [0.00s].
## checking subsets of size 1 2 3 4 5 6 done [0.00s].
## writing ... [9 rule(s)] done [0.00s].
## creating S4 object  ... done [0.00s].

inspect(rules_store)

##     lhs                             rhs                         support confidence  coverage     lift count
## [1] {NumChildren=NoChild}        => {NumStorePurchases=High} 0.12602366  0.4460548 0.2825296 1.531919   277
## [2] {AgeCategory=BabyBoomer,                                                                               
##      NumChildren=NoChild}        => {NumStorePurchases=High} 0.05141037  0.4379845 0.1173794 1.504203   113
## [3] {NumChildren=NoChild,                                                                                  
##      Relationship=Not-Partnered} => {NumStorePurchases=High} 0.05414013  0.4817814 0.1123749 1.654618   119
## [4] {NumChildren=NoChild,                                                                                  
##      Recency=Distant}            => {NumStorePurchases=High} 0.06414923  0.4563107 0.1405823 1.567142   141
## [5] {NumChildren=NoChild,                                                                                  
##      Recency=Recent}             => {NumStorePurchases=High} 0.06187443  0.4358974 0.1419472 1.497035   136
## [6] {NumChildren=NoChild,                                                                                  
##      Education=Graduates}        => {NumStorePurchases=High} 0.06187443  0.4317460 0.1433121 1.482778   136
## [7] {NumChildren=NoChild,                                                                                  
##      Relationship=Partnered}     => {NumStorePurchases=High} 0.07188353  0.4224599 0.1701547 1.450886   158
## [8] {NumChildren=NoChild,                                                                                  
##      YearsJoined=<8 years}       => {NumStorePurchases=High} 0.09463148  0.4342380 0.2179254 1.491336   208
## [9] {NumChildren=NoChild,                                                                                  
##      Relationship=Partnered,                                                                               
##      YearsJoined=<8 years}       => {NumStorePurchases=High} 0.05595996  0.4226804 0.1323931 1.451643   123

3. Catalog sales channel

CustomersAssoc3 <- subset(CustomersAssoc1, select = -NumWebPurchases)
CustomersAssoc3$NumCatalogPurchases <- Customers_cleaned$NumCatalogPurchases

CustomersAssoc3$NumCatalogPurchases <- cut(
  CustomersAssoc3$NumCatalogPurchases,
  breaks = quantile(CustomersAssoc3$NumCatalogPurchases, c(0, 0.33, 0.66, 1)),
  labels = c("Low", "Medium", "High"),
  include.lowest = TRUE
)

rules_catalog <- apriori(CustomersAssoc3, parameter = list(support = 0.05, confidence = 0.6), appearance = list(rhs = "NumCatalogPurchases=High"))

## Warning: Column(s) 1, 2, 3, 4 not logical or factor. Applying default
## discretization (see '? discretizeDF').

## Apriori
## 
## Parameter specification:
##  confidence minval smax arem  aval originalSupport maxtime support minlen
##         0.6    0.1    1 none FALSE            TRUE       5    0.05      1
##  maxlen target  ext
##      10  rules TRUE
## 
## Algorithmic control:
##  filter tree heap memopt load sort verbose
##     0.1 TRUE TRUE  FALSE TRUE    2    TRUE
## 
## Absolute minimum support count: 109 
## 
## set item appearances ...[1 item(s)] done [0.00s].
## set transactions ...[18 item(s), 2198 transaction(s)] done [0.00s].
## sorting and recoding items ... [18 item(s)] done [0.00s].
## creating transaction tree ... done [0.00s].
## checking subsets of size 1 2 3 4 5 6 done [0.00s].
## writing ... [19 rule(s)] done [0.00s].
## creating S4 object  ... done [0.00s].

inspect(rules_catalog)

##      lhs                              rhs                           support confidence   coverage     lift count
## [1]  {NumChildren=NoChild}         => {NumCatalogPurchases=High} 0.18471338  0.6537842 0.28252957 2.103979   406
## [2]  {AgeCategory=BabyBoomer,                                                                                   
##       NumChildren=NoChild}         => {NumCatalogPurchases=High} 0.08052775  0.6860465 0.11737944 2.207804   177
## [3]  {NumChildren=NoChild,                                                                                      
##       Relationship=Not-Partnered}  => {NumCatalogPurchases=High} 0.07233849  0.6437247 0.11237489 2.071606   159
## [4]  {AgeCategory=GenX,                                                                                         
##       NumChildren=NoChild}         => {NumCatalogPurchases=High} 0.05823476  0.6701571 0.08689718 2.156669   128
## [5]  {NumChildren=NoChild,                                                                                      
##       Recency=Distant}             => {NumCatalogPurchases=High} 0.09508644  0.6763754 0.14058235 2.176681   209
## [6]  {NumChildren=NoChild,                                                                                      
##       Recency=Recent}              => {NumCatalogPurchases=High} 0.08962693  0.6314103 0.14194722 2.031976   197
## [7]  {NumChildren=NoChild,                                                                                      
##       Education=Graduates}         => {NumCatalogPurchases=High} 0.09554140  0.6666667 0.14331210 2.145437   210
## [8]  {NumChildren=NoChild,                                                                                      
##       Relationship=Partnered}      => {NumCatalogPurchases=High} 0.11237489  0.6604278 0.17015469 2.125359   247
## [9]  {NumChildren=NoChild,                                                                                      
##       YearsJoined=<8 years}        => {NumCatalogPurchases=High} 0.14285714  0.6555324 0.21792539 2.109605   314
## [10] {AgeCategory=BabyBoomer,                                                                                   
##       NumChildren=NoChild,                                                                                      
##       YearsJoined=<8 years}        => {NumCatalogPurchases=High} 0.06505914  0.7009804 0.09281165 2.255864   143
## [11] {NumChildren=NoChild,                                                                                      
##       Relationship=Not-Partnered,                                                                               
##       YearsJoined=<8 years}        => {NumCatalogPurchases=High} 0.05414013  0.6329787 0.08553230 2.037024   119
## [12] {NumChildren=NoChild,                                                                                      
##       Education=Graduates,                                                                                      
##       Recency=Distant}             => {NumCatalogPurchases=High} 0.05095541  0.6913580 0.07370337 2.224897   112
## [13] {NumChildren=NoChild,                                                                                      
##       Relationship=Partnered,                                                                                   
##       Recency=Distant}             => {NumCatalogPurchases=High} 0.05505005  0.6875000 0.08007279 2.212482   121
## [14] {NumChildren=NoChild,                                                                                      
##       Recency=Distant,                                                                                          
##       YearsJoined=<8 years}        => {NumCatalogPurchases=High} 0.07324841  0.6880342 0.10646042 2.214201   161
## [15] {NumChildren=NoChild,                                                                                      
##       Relationship=Partnered,                                                                                   
##       Recency=Recent}              => {NumCatalogPurchases=High} 0.05732484  0.6363636 0.09008189 2.047917   126
## [16] {NumChildren=NoChild,                                                                                      
##       Recency=Recent,                                                                                           
##       YearsJoined=<8 years}        => {NumCatalogPurchases=High} 0.06960874  0.6244898 0.11146497 2.009705   153
## [17] {NumChildren=NoChild,                                                                                      
##       Education=Graduates,                                                                                      
##       Relationship=Partnered}      => {NumCatalogPurchases=High} 0.05868972  0.6861702 0.08553230 2.208202   129
## [18] {NumChildren=NoChild,                                                                                      
##       Education=Graduates,                                                                                      
##       YearsJoined=<8 years}        => {NumCatalogPurchases=High} 0.07734304  0.6772908 0.11419472 2.179627   170
## [19] {NumChildren=NoChild,                                                                                      
##       Relationship=Partnered,                                                                                   
##       YearsJoined=<8 years}        => {NumCatalogPurchases=High} 0.08871702  0.6701031 0.13239308 2.156496   195